#  $\zeta$-functions and the topology of superlevel sets of stochastic
processes

Daniel Perez Email<EMAIL_ADDRESS>Département de mathématiques et
applications, École normale supérieure, CNRS, PSL University, 75005 Paris,
France Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, CNRS,
91405 Orsay, France  DataShape, Centre Inria Saclay, 91120 Palaiseau, France

###### Abstract

We describe the topology of superlevel sets of ($\alpha$-stable) Lévy
processes X by introducing so-called stochastic $\zeta$-functions, which are
defined in terms of the widely used $\operatorname{Pers}_{p}$-functional in
the theory of persistence modules. The latter share many of the properties
commonly attributed to $\zeta$-functions in analytic number theory, among
others, we show that for $\alpha$-stable processes, these (tail)
$\zeta$-functions always admit a meromorphic extension to the entire complex
plane with a single pole at $\alpha$, of known residue and that the analytic
properties of these $\zeta$-functions are related to the asymptotic expansion
of a dual variable, which counts the number of variations of X of size
$\geq\varepsilon$. Finally, using these results, we devise a new statistical
parameter test using the topology of these superlevel sets. We further develop
an analogous theory, whereby we consider the dual variable to be the number of
points in the persistence diagram inside the rectangle
$]\infty,x]\times[x+\varepsilon,\infty[$.

Index terms— Dirichlet series; zeta functions; stochastic processes; topology
of superlevel sets; persistent homology; self-similar processes; Lévy
processes

MSC2020 Classification— 60D05, 62F03, 30B50, 55N31, 60G18

###### Contents

  1. 1 Introduction
    1. 1.1 Our contribution
  2. 2 Generalities
    1. 2.1 The Mellin transform
      1. 2.1.1 Analytic continuation
    2. 2.2 Connected components of superlevel sets of stochastic processes
    3. 2.3 A crashcourse in persistent homology
      1. 2.3.1 Homology
      2. 2.3.2 Persistent homology
      3. 2.3.3 Persistent homology as a measure
    4. 2.4 Trees and barcodes
    5. 2.5 Integration on trees and the duality between $N^{\varepsilon}$ and $\operatorname{Pers}_{p}^{p}$
      1. 2.5.1 Calculation of $N^{\varepsilon}$ in dimension one
    6. 2.6 $\zeta$-functions associated to stochastic processes
    7. 2.7 Average diagrams and characterizations of Wasserstein convergence of diagrams
  3. 3 $\zeta$-functions of Lévy processes and semimartingales
    1. 3.1 Semimartingales
    2. 3.2 Renewal theory
    3. 3.3 Lévy processes
      1. 3.3.1 Exponential corrections
      2. 3.3.2 Distribution of the length of branches
      3. 3.3.3 Statistical parameter testing for $\alpha$-stable processes and perspectives
    4. 3.4 Propagators and local $\zeta$-functions
      1. 3.4.1 Itô diffusions
      2. 3.4.2 Obtention of the local time for continuous semimartingales
  4. 4 Examples of applications
    1. 4.1 Brownian motion
      1. 4.1.1 Associated $\zeta$-function and asymptotic expansions for $N^{\varepsilon}$
      2. 4.1.2 Distribution of the length of the $k$th longest branch
      3. 4.1.3 Local $\zeta$-functions and corrections to the local time
      4. 4.1.4 Calculation of the average persistence diagram
    2. 4.2 Reflected Brownian motion
      1. 4.2.1 Associated $\zeta$-function and $N^{\varepsilon}$
      2. 4.2.2 Local $\zeta$-function, $N^{x,x+\varepsilon}$ and local time
    3. 4.3 Brownian motion with drift
      1. 4.3.1 Distribution of $N^{x,x+\varepsilon}_{B^{\mu,\sigma}}$ and the local time $L_{t}^{x}(B^{\mu,\sigma})$
    4. 4.4 Ornstein-Uhlenbeck process
  5. 5 Acknowledgements

##  1 Introduction

The problem of the characterization of the topology of superlevel sets of
random functions has been a long studied topic in the theory of random fields.
While a complete description has been thus far unknown, partial descriptors of
the topology of superlevel sets, such as their Euler characteristic, have been
described for certain classes of random processes [2, 3, 4, 37, 13, 32]. Thus
far, the study of the homology of superlevel sets of random functions in
dimension one has focused on either smooth random (Gaussian) fields [2, 3], or
irregular processes which are in some sense canonical, such as Brownian motion
[13, 37, 4]. In this paper, following the universality reasoning detailed in
[36, §3], we will adopt the second point of view while enlarging the category
of processes considered to objects acting as universal limits of random
processes in 1D.

This work is another stage in a program started in [35] and later continued in
[36], which aimed to characterize the barcodes of random functions as
completely as possible (in dimension one). To do this, we adopted the tree
formalism originally developped by Le Gall [19, 20], which brings benefits in
the probabilistic setting. This formalism allowed us to partially study the
case of Markov or self-similar processes, and to processes admitting the two
latter as limits [36]. In this paper, we further develop the theory to
describe almost completely the case of ($\alpha$-stable) Lévy processes.

The understanding of so-called topological noise is an active area of research
in Topological Data Analysis (TDA) (cf. section 2.3 for a quick introduction,
or [12, 34] for a more comprehensive one). This topological noise is
characterized by the behaviour of the small bars of the barcode of a function
and its role is particularly difficult to grasp. Nonetheless, topological
noise has proved useful in a variety of different applications, which seem to
exploit the information contained therein, sometimes directly, or indirectly
through the use of Wasserstein $p$ metrics on the space or functionals such as
$\operatorname{Pers}_{p}$ [15, 31, 43, 17, 10], despite the absence of general
stability results. The program described above is based around the intuition
that while it might be hopeless to find a general stability theorem, there
should be a form of statistical stability of barcodes.

By statistical stability we mean that any two samples drawn from the same
probability distribution should have “close” or “similar” behaviour of its
topological noise. For example, this paper shows that, at least in the simple
setting of 1D and for a fairly wide range of distributions, this topological
noise does indeed exhibit the robustness sought. More precisely, we show that
the number of bars of length $\geq\varepsilon$, $N^{\varepsilon}$, is a dual
to $\operatorname{Pers}_{p}^{p}$, and that it exhibits the statistical
robustness required, such as an almost sure asymptotic behaviour as
$\varepsilon\to 0$.

If this notion of statistical stability is a good one, a natural subsequent
question is whether it is possible to differentiate stochastic processes given
their topological signature. In this paper, we partially answer this question
positively through the development of a statistical test constructed with the
functional $N^{\varepsilon}$, which can differentiate $\alpha$-stable
processes for different values of $\alpha$. In dimension one, while
interesting, this development is unlikely to do better than wavelet analysis
or other known techniques (cf. [16] and the references therein). However,
further developments in this direction could eventually lead to “topological
statistics”, i.e. robust statistical tests for random fields, for which all
known techniques do not generalize, but for which barcodes are easily
computable.

This discussion and the results of this paper hint at the fact that
statistical stability is a correct notion of stability to consider, and there
are many open questions to be tackled, some of these questions are:

  * •

Dimensionality: are the statistical stability results of this paper particular
to dimension one, or do they generalize in some way to higher dimensional
random fields?

  * •

Signal vs. noise problem: Given the regular structure of topological noise
shown in this paper, does this allow us to detect the presence of an
underlying topological signal?

  * •

Statistical robustness of topological noise: What can we say, quantitatively,
about the variation of topological noise induced by perturbations of the
distribution of the noise (for instance in some Wasserstein metric)?

  * •

Best proxys for topological signatures: much of this paper was inspired by
what is used in practice, namely Wasserstein $p$ metrics and the
$\operatorname{Pers}_{p}$ functional and its dual $N^{\varepsilon}$. However,
there is no guarantee this yields the best possible proxys to answer the two
previous problems. In this regard, is it possible to prove or disprove which
proxys do best in what context?

From a more probabilistic point of view, this paper introduces so-called
$\zeta$-functions associated to a stochastic process, constructed using the
$\operatorname{Pers}_{p}^{p}$ functional we previously discussed. The main,
and perhaps most important, departure from the conventional TDA theory is that
we will consider this quantity for complex $p$ for reasons which will become
evident throughout this paper, but which are analogous to the ones behind the
complexification of the Riemann $\zeta$-function in analytic number theory.
There are some similarities to the results of Pitman, Yor and Biane in [39, 6,
38] regarding the probabilistic interpretation of the $\zeta$-function and
more generally $L$-functions based on connections with some families of
infinitely divisible distributions connected to Brownian motion. However, the
$\zeta$-functions herein are of different nature to those considered by
Pitman, Yor and Biane, as they stem from a different construction. For
Brownian motion, we fall back on one of the infinitely divisible distributions
considered by these authors. Renewal theory [23] intervenes at many different
steps in this paper and combined with the results of Pitman, Yor and Biane, it
is a posteriori perhaps not surprising that the $\zeta$-functions hereby
introduced share some of the analycity properties of the Riemann
$\zeta$-function.

###  1.1 Our contribution

More precisely, our contribution can be split along the following lines:

  1. 1.

We establish a duality relation with respect to the Mellin transform between
the study of $\operatorname{Pers}_{p}^{p}$ and the number of leaves of a
$\varepsilon$-trimmed tree $\geq\varepsilon$, $N^{\varepsilon}$ (cf. section
2.5). With the help of a correct notion of integration on trees developped in
[37], it is possible to prove an interpolation theorem for
$\operatorname{Pers}_{p}^{p}$ (proposition 2.18);

  2. 2.

We introduce $\zeta$-functions for stochastic processes and for persistent
measures (cf. section 2.6 and section 2.7 respectively). We show an
interpolation theorem for Wasserstein $p$-distances between diagrams
(proposition 2.41) and a characterization of convergence between diagrams in
these Wasserstein distances in terms of the $\zeta$-functions introduced
(theorem 2.45).

  3. 3.

We show that in the context of $\alpha$-stable Lévy processes, the associated
(tail) $\zeta$-functions always admit a meromorphic extension to the entire
complex plane, with a unique pole at $p=\alpha$ with known residue (theorem
3.25). By duality, this meromorphic extension implies the existence of an
asymptotic series for $N^{\varepsilon}$ as $\varepsilon\to 0$, which we
explicitly calculate up to superpolynomial (i.e. smaller than any polynomial)
corrections (theorem 3.15). An explicit form of the meromorphic continuation
of $\hat{\zeta}$ is shown to be related to the superpolynomial corrections to
the asymptotic expansion of theorem 3.15 (cf. section 2.1.1). We also define a
generating function for the length of the $k$th longest bar (cf. section
3.3.2);

  4. 4.

We give an almost sure result detailing the asymptotics of $N^{\varepsilon}$
for any continuous semimartingale (proposition 3.1), which turns out to be
related to the quadratic variation of the process, providing further evidence
for the link between regularity and $\operatorname{Pers}^{p}_{p}$ hinted at in
[37, 40]. We conclude from this that the $\zeta$-functions associated to
continuous semimartingales always have a simple pole at $p=2$;

  5. 5.

We apply the theory above to different stochastic processes, such as Brownian
motion, reflected Brownian motion. We derive explicit formulæ for the
respective $\zeta$-functions of these processes and infer the associated
asymptotic expansions of $N^{\varepsilon}$ (theorems 4.2, 4.14 propositions
4.4 and 4.15) and in the case of Brownian motion, the explicit distribution of
the length of the $k$th longest bar (cf. section 4.1.2).

  6. 6.

We design a statistical test for the parameter $\alpha$ of $\alpha$-stable
Lévy processes by using the theory previously described (cf. section 3.3.3);

  7. 7.

We study local trees and introduce local $\zeta$-functions (cf. section 2.6)
and deduce formulæ for the number of points contained in the rectangle
$]\\!-\\!\infty,x]\times[x+\varepsilon,\infty[$ of the persistence diagram,
$N^{x,x+\varepsilon}$, by introducing the notion of propagators, which, for
Markov processes, reduces the problem of the study of $N^{x,x+\varepsilon}$ to
the study of hitting times of the process (cf. section 3.4), in particular, we
link the regularity of these propagators with meromorphic extensions of the
local $\zeta$-functions of the process (proposition 3.40);

  8. 8.

Finally, we apply the theory above to different stochastic processes, such as
Brownian motion, reflected Brownian motion, Brownian motion with drift and the
Ornstein-Uhlenbeck process for which explicit computations are possible. We
derive explicit formulæ for the respective (local, global) $\zeta$-functions
of these processes and infer the associated asymptotic expansions of
$N^{x,x+\varepsilon}$ (theorems 4.6, 4.14 propositions 4.13 and 4.15 and
section 4.3). We also infer formulæ regarding the Ornstein-Uhlenbeck process,
in particular concerning its local time (cf. section 4.4).

##  2 Generalities

###  2.1 The Mellin transform

######  Definition 2.1.

Let $f$ be a locally integrable function over the ray $]0,\infty[$. The Mellin
transform of $f$ is

${\mathcal{M}}[f(x)](s):=\int_{0}^{\infty}x^{s-1}f(x)\;dx\,.$

Note that $d\log(x)=\frac{dx}{x}$ is the Haar measure of
$(\mathbb{R}_{+},\times)$. The Mellin transform reflects the Pontryagin
duality with respect to this locally compact abelian group. Its theory is
analogous to that of the bilateral Laplace transform, as the map
$\log:(\mathbb{R}_{+},\times)\to(\mathbb{R},+)$ induces an isomorphism of
abelian groups.

######  Notation 2.2.

For convenience, we will also employ the shorthand notation
${\mathcal{M}}[f](s)=f^{*}(s)$.

######  Definition 2.3.

The fundamental strip of $f$, $\langle\alpha,\beta\rangle$ is the maximal set

$\langle\alpha,\beta\rangle:=\left\\{z\in\mathbb{C}\,|\,\alpha<\operatorname{Re}(z)<\beta\right\\}$

where $f^{*}(s)$ is well defined.

The Mellin transform can be inverted by virtue of the following theorem, which
follows from the Laplace inversion theorem.

######  Theorem 2.4 (Mellin inversion,[18, 33]).

Let $f$ have fundamental strip $\langle\alpha,\beta\rangle$ and let
$c\in\,]\alpha,\beta[$. Then

  1. 1.

If $f$ is integrable and $f^{*}(c+it)$ is integrable, then for almost every
$x$

$f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}f^{*}(s)x^{-s}\;ds$

If $f$ is continuous, the equality holds everywhere.

  2. 2.

If $f$ is locally integrable and of bounded variation in a neighbourhood of
$x$, then

$\frac{f(x^{+})+f(x^{-})}{2}=\lim_{T\to\infty}\frac{1}{2\pi
i}\int_{c-iT}^{c+iT}f^{*}(s)x^{-s}\;ds$

A sufficient condition for the Mellin transform to be well-defined on
$\langle\alpha,\beta\rangle$ is that the function is such that

$f(x)=O(x^{-\alpha})\;\text{as }x\to 0\quad\text{and}\quad
f(x)=O(x^{-\beta})\;\text{as }x\to\infty\,.$

In fact, Mellin transforms are a good tool to study asymptotic expansions as
suggested by the following theorem.

Figure 1: Contour for the evaluation of the Bromwich integral of the inverse
Mellin transform.

######  Theorem 2.5 (Fundamental correspondence, [21]).

Let $f:\;]0,\infty[\to\mathbb{C}$ be a continuous function with non-empty
fundamental strip $\langle\alpha,\beta\rangle$. Then,

  * •

Assume that $f^{*}(s)$ admits a meromorphic continuation to the strip
$\langle\gamma,\beta\rangle$ for $\gamma<\alpha$, that it has only a finite
amount of poles there and that it is analytic on
$\operatorname{Re}(s)=\gamma$. Assume also that there exists
$\eta\in\,]\alpha,\beta[$ such that along a denumerable set of horizontal
segments with $\left\lvert\operatorname{Im}(s)\right\rvert=T_{i}$ where
$T_{i}\to\infty$,

$f^{*}(s)=O(\left\lvert s\right\rvert^{-r})\text{ with }r>1\text{ as
}\left\lvert s\right\rvert\to\infty\text{ and }s
\in\langle\gamma,\eta\rangle\,.$

Indexing the poles on $\langle\gamma,\beta\rangle$ by their location $\xi$ and
by their order $k$ and denoting $c_{\xi,k}$ the $k$th coefficient in the
Laurent expansion around $\xi$ of $f^{*}(s)$, we have an asymptotic expansion
of $f$ around $0$

$f(x)\sim\sum_{(\xi,k)}c_{\xi,k}\frac{(-1)^{k-1}}{(k-1)!}x^{-\xi}\log^{k}(x)+O(x^{-\gamma})\quad\text{as
}x\to 0\,.$

  * •

Conversely, if the function $f$ has such an asymptotic expansion around $0$,
then $f^{*}(s)$ has a meromorphic continuation to the strip
$\langle\gamma,\beta\rangle$.

Furthermore, an analogous statement holds true for asymptotic expansions
around $\infty$ and meromorphic continuations beyond $\beta$.

###### Sketch of proof.

It suffices to perform contour integration using the contour of figure 1. The
estimates of the theorem allow us to discard the top and bottom integrals and
to state that the integral of the path along $\operatorname{Re}(p)=\gamma$ is
$O(x^{-\gamma})$. Conversely, consider

$f(x)\sim\sum_{(\xi,k)}c_{\xi,k}x^{\xi}\log^{k}(x)+O(x^{-\gamma})\quad\text{as
}x\to 0$

for some $\gamma<\alpha$. It follows that

$\displaystyle f^{*}(s)$
$\displaystyle=\sum_{(\xi,k)}c_{\xi,k}\frac{(-1)^{k}k!}{(s+\xi)^{k+1}}+\int_{1}^{\infty}x^{s-1}f(x)\;dx$
$\displaystyle+\int_{0}^{1}x^{s-1}\underbrace{\left[f(x)-\sum_{(\xi,k)}c_{\xi,k}
x^{\xi}\log^{k}(x)\right]}_{=O(x^{-\gamma})}\;dx\,,$

which is well-defined on the strip $\langle\gamma,\beta\rangle$. ∎

$f(x)$ | $f^{*}(s)$ | $\langle\alpha,\beta\rangle$
---|---|---
$x^{\nu}f(x)$ | $f^{*}(s+\nu)$ | $\langle\alpha-\nu,\beta-\nu\rangle$
$f(x^{\nu})$ | $\frac{1}{\nu}f^{*}(\tfrac{s}{\nu})$ | $\langle\nu\alpha,\nu\beta\rangle$
$f(x^{-1})$ | $f^{*}(-s)$ | $\langle-\beta,-\alpha\rangle$
$f(\lambda x)$ | $\lambda^{-s}f^{*}(s)$ | $\langle\alpha,\beta\rangle$
$\frac{\partial}{\partial x}f(x)$ | $-(s-1)f^{*}(s-1)$ |
$\int_{0}^{x}f(t)\;dt$ | $-\frac{1}{s}f^{*}(s+1)$ |
Table 1: Functional properties of the Mellin transform $f(x)$ | $f^{*}(s)$ | $\langle\alpha,\beta\rangle$
---|---|---
$e^{-x}$ | $\Gamma(s)$ | $\langle 0,\infty\rangle$
$e^{-x^{2}}$ | $\frac{1}{2}\Gamma(\frac{s}{2})$ | $\langle 0,\infty\rangle$
$\operatorname{erfc}(x)$ | $2^{-s}\frac{\Gamma(s)}{\Gamma(1+\frac{s}{2})}$ | $\langle 0,\infty\rangle$
$\operatorname{csch}(x)$ | $2^{1-s}\left(2^{s}-1\right)\Gamma(s)\zeta(s)$ | $\langle 1,\infty\rangle$
$\operatorname{csch}^{2}(x)$ | $2^{2-s}\Gamma(s)\zeta(s-1)$ | $\langle 2,\infty\rangle$
$\frac{1}{e^{x}-1}$ | $\Gamma(s)\zeta(s)$ | $\langle 1,\infty\rangle$
Table 2: A short dictionary of Mellin transforms.

####  2.1.1 Analytic continuation

As stated by the fundamental correspondence (theorem 2.5), the existence of an
asymptotic expansion around $0$ of $f(x)$ entails a meromorphic continuation
of $f^{*}(s)$ to a larger strip. If $f(x)$ admits a converging Laurent series
(with finite singular part) on some open disk around the origin, then this
extension is in fact valid over all of $\mathbb{C}$, and the residues of the
poles of $f^{*}(s)$ will be related to the Laurent coefficients of $f(x)$. It
turns out that in this context, one can even write an explicit integral
representation for the extension of $f^{*}(s)$.

######  Lemma 2.6 (Integral representation of $f^{*}$).

Let $f$ be a meromorphic function admitting a Laurent series at $0$, with
singular part of degree $n$, holomorphic on a neighbourhood of
$\mathbb{R}_{+}^{*}$ and integrable over the Hankel contour (cf. figure 2).
Suppose further that its fundamental strip $\langle n,\beta\rangle$ is non-
empty. Then, the function $f^{*}$ admits a meromorphic continuation on
$\langle-\infty,\beta\rangle$ given by

$\displaystyle f^{*}(s)$ $\displaystyle=\frac{e^{-i\pi s}}{2i\sin(\pi
s)}\oint_{H}z^{s-1}f(z)\;dz$ $\displaystyle=-\frac{\Gamma(s)\Gamma(1-s)}{2\pi
i}\oint_{H}(-z)^{s-1}f(z)\;dz\,,$

where $H$ denotes the Hankel contour.

Figure 2: The Hankel contour $H$.

###### Proof.

We start by splitting the Hankel contour into three pieces.

  1. 1.

A segment from $\infty+i\varepsilon$ to $\nu+i\varepsilon$;

  2. 2.

A circle $C_{\nu}$ around the origin of radius $\nu$;

  3. 3.

A segment from $\nu-i\varepsilon$ to $\infty-i\varepsilon$.

For $s\in\langle n,\beta\rangle$, $f$ is holomorphic everywhere on this
contour, so that we may take $\varepsilon=0$ according to Cauchy’s theorem.
Notice also that

$\int_{C_{\nu}}z^{s-1}B(z)\;dz=O(\nu^{\operatorname{Re}(s)-n})\to
0\quad\text{as }\nu\to 0\,.$

It follows that for $\operatorname{Re}(s)>n$

$\displaystyle\oint_{H}z^{s-1}f(z)\;dz$ $\displaystyle=\lim_{\nu\to
0}\left\\{\int_{\infty}^{\nu}+\int_{C_{\nu}}+\int_{\nu e^{2\pi i}}^{\infty
e^{2\pi i}}\right\\}z^{s-1}f(z)\;dz$ $\displaystyle=(e^{2\pi
i(s-1)}-1)\int_{0}^{\infty}z^{s-1}f(z)\;dz$ $\displaystyle=2ie^{i\pi
s}\sin(\pi s)f^{*}(s)\,,$

as desired. The integral over the complex contour $H$ converges for all
$s\in\langle-\infty,\beta\rangle\setminus\\{n\\}$. Finally, the second
expression for $f^{*}$ is obtained through Euler’s reflection formula, namely

$\Gamma(p)\Gamma(1-p)=\frac{\pi}{\sin(\pi p)}\,,$

which after some simplification yields the desired expression. ∎

######  Remark 2.7.

If $f$ has fundamental strip $\langle n,\infty\rangle$, then the extension
given by this procedure holds over $\mathbb{C}$.

Furthermore, if $f$ posseses a meromorphic continuation to
$\mathbb{C}\setminus\mathbb{R}_{+}$ (i.e. we admit the possibility of a branch
cut on the positive real axis), then we can find a more explicit formulation
for the Hankel representation of $f^{*}$.

######  Lemma 2.8 (Functional equation of $f^{*}$).

Suppose $f$ posseses a meromorphic continuation to
$\mathbb{C}\setminus\mathbb{R}_{+}$ and denote $\mathcal{P}$ the set of poles
of $f$ not including $0$. Suppose further that $f$ has the following decay
condition : for all $s\in\langle n,\beta\rangle$ and for some monotone
increasing sequence of radii $r_{n}\to\infty$ as $n\to\infty$,

$\int_{C_{r_{n},\varepsilon}}\left\lvert
z^{s-1}f(z)\right\rvert\;dz\xrightarrow[n\to\infty]{}0\,.$

where $C_{r_{n},\varepsilon}$ is the circle of radius $r_{n}$ minus a small
(symmetric) arc of length $\varepsilon$ around the positive real axis (cf.
figure 3). Then,

$f^{*}(s)=\Gamma(s)\Gamma(1-s)\sum_{z_{0}\in\mathcal{P}}\operatorname{Res}((-z)^{s-1}f(z);z_{0})\,.$

###### Proof.

The proof relies on the use of the residue theorem by completing the Hankel
contour into a Pac-Man (cf. figure 3), whose circular contribution is going to
zero, due to the assumption of the lemma.

Figure 3: The Pac-Man contour.

By the residue theorem, we then have

$\displaystyle
f^{*}(s)=\Gamma(s)\Gamma(1-s)\sum_{z_{0}\in\mathcal{P}}\operatorname{Res}((-z)^{s-1}f(z);z_{0})\,,$

as desired. ∎

###  2.2 Connected components of superlevel sets of stochastic processes

Let us briefly recall the construction of a tree from a continuous function
$f:[0,1]\to\mathbb{R}$. For a more complete description of this, the reader is
welcome to consult [19, 35].

######  Definition/Proposition 2.9 ([19]).

Let $x<y\in[0,1]$, the function

$d_{f}(x,y):=f(x)+f(y)-2\min_{t\in[x,y]}f(t)$

is a pseudo-distance on $[0,1]$ and the quotient metric space

$T_{f}:=[0,1]/\\{d_{f}=0\\}$

with distance $d_{f}$ is a rooted $\mathbb{R}$-tree, whose root coincides with
the image in $T_{f}$ of the point in $[0,1]$ at which $f$ achieves its
infimum.

Figure 4: A function $f$ and its associated tree $T_{f}$ in red.

The tree $T_{f}$ has the particularity that its branches correspond to
connected components of the superlevel sets of $f$, as illustrated by figure
4. Let us now introduce the so-called $\varepsilon$-simplified or
$\varepsilon$-trimmed tree of $T_{f}^{\varepsilon}$. This object is obtained
by “giving a haircut” of length $\varepsilon$ to $T_{f}$. More precisely, if
we define a function $h:T_{f}\to\mathbb{R}$ which to a point $\tau\in T_{f}$
associates the distance from $\tau$ to the highest leaf above $\tau$ with
respect to the filtration on $T_{f}$ induced by $f$, then

######  Definition 2.10.

Let $\varepsilon\geq 0$. An $\varepsilon$-trimming or
$\varepsilon$-simplification of $T_{f}$ is the metric subspace of $T_{f}$
defined by

$T_{f}^{\varepsilon}:=\\{\tau\in T_{f}\,|\,h(\tau)\geq\varepsilon\\}$

######  Notation 2.11.

Let us denote $N^{\varepsilon}$ the number of leaves of $T_{f}^{\varepsilon}$.

###  2.3 A crashcourse in persistent homology

Throughout this section, we will detail and give the ideas behind persistent
homology. A proper introduction to this is out of the scope of this paper, so
we encourage the reader to consult the following classical references about
this topic [24, 30, 12, 34]. This section aims nonetheless to give a brief
introduction compiling the main results and intuition behind this field. To do
so, it is convenient to break down the topic along its title. First, we will
briefly recall what homology is and how it can be defined, and then we will
explain the persistent aspect of persistent homology. It goes without saying
that a reader familiar with these concepts may skip this section entirely.

####  2.3.1 Homology

In general, the motivation behind the introduction of objects in algebraic
topology such as homology is to study topological spaces through algebra. That
is, to attach an algebraic object (such as a module, a group, etc.) to a
topological space, in such a way that, loosely speaking, this algebraic object
remains invariant for any two homeomorphic topological spaces. Furthermore, we
would like this invariant to behave well with respect to continuous maps.
Namely, if we have a continuous map between two topological spaces $f:X\to Y$,
we would like to have an induced morphism at the level of the two invariants
we attached to $M$ and $Y$. The most famous such invariant for topological
spaces is the fundamental group $\pi_{1}(X)$ first introduced by Poincaré.
Some useful references for further reading are [24, 30].

In the terms of category theory, the above discussion is equivalent to saying
that we use functors between the category of topological spaces, Top, and a
category of algebraic objects, such as the category of groups, Grp, or that of
modules over some ring $R$, $\textbf{Mod}_{R}$. In this sense, homology is
nothing other than a functor $H_{*}:\textbf{Top}\to\textbf{Mod}_{R}$. For our
purposes, it is sufficient to consider the ring $R$ to be a field $k$, so that
we are really working over the category of vector spaces over this field
$\textbf{Vect}_{k}$. We will not detail the precise definitions of these
objects, as we will not really need them, but a good reference as an
introduction to category theory is given by Mac Lane in [27].

Recall that finding such a functor entails attaching a vector space to a
topological space, in such a way that continuous maps between topological
spaces induce linear maps at the level of vector spaces. To render this
practical, let us first focus on triangulable spaces. Namely, spaces which are
homeomorphic to a simplicial complex (a set of oriented simplices glued to
one-another along edges, $n$-faces or points). Given a simplicial complex $M$,
we can define a so-called chain complex, which is nothing other than a
sequence of vector spaces, denoted $C_{*}(M,k)$, called the space of chains
(the star denotes an index, which we call the degree), where $C_{n}$ is the
free $k$-vector space generated by the set of $n$-faces in the simplicial
complex. For the sake of notational simplicity, whenever the simplicial
complex we are talking about and the field over which we are working on is
clear, we may drop $M$ and $k$ and denote $C_{*}(M,k)=C_{*}$.

We can define a linear map $\partial:C_{*}\to C_{*}$ called the boundary map,
which is defined degree-by-degree as follows. The boundary map $\partial$
sends the generator of an $n$-face (an element of $C_{n}$) to the (signed) sum
of the generators of its boundary (which are elements of $C_{n-1}$), where the
sign in front of each generator is determined by the compatibility of its
orientation with the orientation of the $n$-face. With this definition
$\partial^{2}=0$, which reflects the fact that the boundary of a boundary is
always empty. So, we can see a chain complex as some graded vector space
$C_{*}$ along with the map $\partial$. Looking at the restriction of
$\partial$ to each degree of $C_{*}$, we can write $\partial$ as a chain of
morphisms

$\cdots\to
C_{2}\xrightarrow{\partial}C_{1}\xrightarrow{\partial}C_{0}\xrightarrow{\partial}0\,,$

with the property that $\partial^{2}=0$. This property implies in particular
that $\operatorname{Im}(\partial)\subset\ker(\partial)$. We call
$\ker(\partial)$ the space of cycles, reflecting the fact that elements of
$\ker(\partial)$ tend to be “loops” or “cycles” of $n$-chains. Homology
quantifies exactly which cycles are not boundaries. More precisely, fixing a
degree $n$, we can define the $n$th homology group over the field $k$ as

$H_{n}(M,k)=\ker(\partial|_{C_{n}})/\operatorname{Im}(\partial|_{C_{n+1}})\,,$

where $\partial|_{C_{n}}$ denotes the restriction of $\partial$ to $C_{n}$. In
some sense, this gives a definition of what we mean by an $n$-dimensional
hole. All of this discussion is best illustrated by an example. In order to
avoid sign problems, it is often practical to work over $k=\mathbb{Z}_{2}$, so
as to simplify calculations. In general, this is restrictive, but it is enough
for our purposes and let us fix the simplicial complex $M$ of figure 5.

Figure 5: An example of a simplicial complex $M$.

In this case, we see that $C_{0}=\langle A,B,C\rangle_{\mathbb{Z}_{2}}$ and
$C_{1}=\langle a,b,c\rangle_{\mathbb{Z}_{2}}$ with all of the higher chains
being $0$. Furthermore, the boundary operator sends

$\partial:a\mapsto B+C\;,\;b\mapsto A+C\;,\;c\mapsto A+B\,,$

with all other generators being sent to zero. From this, it is clear that
$\ker(\partial|_{C_{1}})=(a+b+c)\mathbb{Z}_{2}$, which represents the cycle
which loops around the simplicial complex and $\ker(\partial|_{C_{0}})=C_{0}$.
It follows that $H_{1}(X)=(a+b+c)\mathbb{Z}_{2}$, so that we indeed detect a
hole inside the complex. If we now fill that hole with a cell which fills the
triangle (let us note it $\Delta$), we now have
$C_{2}=\langle\Delta\rangle_{\mathbb{Z}_{2}}$, and its boundary
$\partial(\Delta)=a+b+c$, so that by filling this “hole”, we have effectively
killed the $H_{1}$. As for $H_{0}$, it always quantifies the number of
connected components, as (finite) simplicial complexes are connected if and
only if they are path connected, so these paths constitute cycles which map
any two elements of $C_{0}$ to each other. In particular, if we suppose that
$X$ is path connected, then $\operatorname{Im}(\partial|_{C_{1}})$ is always
generated by all the sums of pairs of vertices in $C_{0}$, so we get one
generator of $H_{0}$ for every connected component.

At this point, given some triangulable space $Y$, the reader might be worried
whether this definition depends on the triangulation we chose for $Y$, but it
is a theorem that this is a well-defined invariant of topological spaces, cf.
[24] for details. In fact, we have adopted a rather restrictive point of view
throughout this discussion, as in reality we can make sense on how homology
can be defined in more general settings [30].

####  2.3.2 Persistent homology

Now that we have roughly sketched out what homology is, let us introduce the
idea behind persistent homology. Once again, we refer the reader to consult
the following references if he or she desires a more detailed description of
the theory [12, 34]. In this case, instead of dealing with a simple chain
complex $C_{*}(M)$, we induce a filtration on this complex, which is typically
done by giving a function on the underlying topological space $M$. For
example, if $M$ is a differentiable manifold and $f:M\to\mathbb{R}$ is a
smooth function, then we can filter the complex $C_{*}(M)$ by considering the
subsets $M_{r}=\\{f>r\\}$ and considering $C_{*}(M_{r})$. We call this
filtration of the complex the superlevel filtration (an analogous definition
can be given for sublevel filtrations). Of course, we may then compute the
homology of $M_{r}$ for every $r$, which gives us a family of vector spaces
indexed by $r$. However, by the functorial nature of $H_{*}$, the fact that we
have a (continuous) inclusion $i_{r,s}:M_{r}\xhookrightarrow{}M_{s}$ for every
$r>s$ yields a linear map between the vector spaces
$H_{*}(i_{r,s}):H_{*}(M_{r})\to H_{*}(M_{s})$.

These morphisms are of interest to us, as they tell us about how the homology
changes as we vary the level $r$. This motivates the study of the so-called
persistent homology, which is nothing other than the family of vector spaces
$(H_{*}(M_{r}))_{r}$ and the family of morphisms $(H_{*}(i_{r,s}))_{r>s}$.
This is more comfortably expressed in the language of category theory :
persistent homology is a functor $H_{*}:\mathbb{R}\to\textbf{Vect}_{k}$, where
$\mathbb{R}$ is seen as a small category (the objects are elements of
$\mathbb{R}$ and there is a morphism between $r\to s$ if and only if $r>s$)
defined by $H_{*}(r)=H_{*}(M_{r})$ and $H_{*}(r\to s)=H_{*}(i_{r,s})$.

If the function $f$ is nice enough, for instance $C^{1}$ and $M$ is compact,
the persistent homology induced by the superlevel filtrations of $f$ can be
decomposed into so-called interval modules. The latter are themselves functors
defined as follows. Fixing a field $k$ and if $A$ is an interval of
$\mathbb{R}$, then

$k_{A}(r):=\begin{cases}k&\text{if }r\in A\\\
0&\text{else}\end{cases}\quad\quad k_{A}(r\to
s)=\begin{cases}\operatorname{id}&\text{if }r,s\in A\\\
0&\text{else}\end{cases}$

The decomposition theorem states the following (cf. Oudot’s book [34] for a
more complete description).

######  Theorem 2.12 (Decomposition theorem, Auslander, Ringel, Tachikawa,
Gabriel, Azumaya).

Under some conditions for $M$ and $f$, if $H_{*}(M,f)$ denotes the persistent
homology with values in $\textbf{Vect}_{k}$, then $H_{*}(M,f)$ it is
isomorphic to a (possibly infinite) direct sum of interval modules. Moreover,
this decomposition is unique up to isomorphism and permutation of the terms.

This theorem entails that if the filtration function $f$ and the space $M$ are
nice enough, the persistent homology functor $H_{*}$ in fact decomposes as a
direct sum of interval modules, more precisely, fixing a degree in homology we
have

$H_{n}=\bigoplus_{i}k_{A_{i}}\,,$

where the $A_{i}$ are intervals of $\mathbb{R}$. Notice then that this means
that the information contained in the persistence module can be encoded by
these intervals $A_{i}$. This collection of intervals is what we call the
barcode associated to a function $f:M\to\mathbb{R}$, typically denoted
$\mathcal{B}(f)$. Another way of representing this information is by keeping
track of the endpoints of the interval. In this way, we may represent the
intervals as a collection of points in the half-plane

$\mathcal{X}:=\\{(x,y)\in(\mathbb{R}\cup\\{\pm\infty\\})^{2}\,|\,x<y\\}\,.$

This collection of points is called the persistence diagram associated to $f$,
and is typically denoted $\operatorname{Dgm}(f)$.

This is not exactly the full story, as there are some technical caveats to
this. Indeed, the theorem requires “nice enough” $M$ and $f$. Throughout this
paper we will be dealing with $C^{0}$ funtions (or in $C^{0}$, up to a finite
number of discontinuities), for which these spaces could be of infinite
dimension. Crucially, however, if $M$ is compact and $f$ is $C^{0}$, the rank
of the maps $H_{*}(i_{r,s})$ is always finite. Under this condition, the
decomposition theorem above applies so we may consider our modules to be
decomposable (cf. [11, 34] for details).

Specializing all of this to $H_{0}$ amounts to talking about connected
components of superlevel sets. In this sense, $\dim H_{0}(M_{r})$ is exactly
equal to the number of connected components of $M_{r}$. The rank of
$H_{0}(i_{r,s})$ corresponds to the number of connected components of $M_{s}$
which contain all of $M_{r}$. The decomposition theorem can also be easily
understood in this setting : bars in the barcode (or equivalently points in
the persistence diagram) indicate when a connected component was “born” and
when it gets absorbed by another one, with the rule that the “eldest”
connected component is the one which always “survives”.

####  2.3.3 Persistent homology as a measure

There is one last point to be discussed, which will come in useful in section
2.7. That is, it can be useful to consider persistence diagrams as measures
over the upper half-plane of $\mathbb{R}^{2}$. For q-tame modules, we can
define such a measure as follows : if $a<b\leq c<d\in\mathbb{R}$, then

$\operatorname{Dgm}(f)([a,b]\times[c,d]):=\operatorname{rank}(H_{*}(i_{b,c}))-\operatorname{rank}(H_{*}(i_{a,c}))+\operatorname{rank}(H_{*}(i_{b,d}))-\operatorname{rank}(H_{*}(i_{a,d}))\,.$

This turns out to be a measure, once we have ironed out some details, as done
in [12]. As an example, if the module is decomposable, then the persistence
measure is nothing other than the sum of Dirac masses at every point of the
persistence diagram. As we will see, considering persistent diagrams as
measures has considerable advantages. The reason for this is because the space
of measures is a linear space, so that we can operate on the space of diagrams
with greater ease than in the algebraic context. This injection onto a linear
space yields some desirable properties for persistence diagrams : for example,
the ability to define Fréchet means [43], or a notion of mean and expectation.
This suggests that this injection is crucial in simplifying the study of
persistence diagrams in a probabilistic setting.

Furthermore, the fact that it is a space of measures allows us to use tools,
such as optimal transport. In particular, there are natural notions of
distances – so-called Wasserstein distances – which stem from this point of
view. This approach has been explored in [17], to which we refer the reader
for further details. The idea is to use the diagonal as a source of infinite
mass, and study the optimal transport distances between diagrams (now seen as
measures), where the distance on the base space (the upper-half plane) is the
$\ell^{\infty}$-distance on the plane, namely

$d_{\mathbb{R}^{2},\infty}((p,q),(r,s))=\max\\{\left\lvert
p-r\right\rvert,\left\lvert q-s\right\rvert\\}\,.$

######  Definition 2.13 (Wasserstein $p$ distances for diagrams).

If $\mu,\nu$ are two persistent measures (to which we have added the diagonal
$\Delta$ as a source of infinite mass), then the Wasserstein $p$ distance
between two diagrams is given by

$d_{p}(\mu,\nu):=\left[\inf_{\pi\in\Gamma(\mu,\nu)}\int_{\overline{\mathcal{X}}^{2}}\\!\\!\\!d_{\mathbb{R}^{2},\infty}^{p}(x,y)\;d\pi(x,y)\right]^{1/p}$

where $\Gamma(\mu,\nu)$ denotes the set of all measures on
$\overline{\mathcal{X}}^{2}$ whose marginals are $\mu$ and $\nu$.

If $p=\infty$ we take the $\sup$-norm, which is exactly the so-called
bottleneck distance referred to in [34, 12], which we may define as follows,
for any two persistent measures $\mu,\nu$,

$d_{\infty}(\mu,\nu):=\inf_{\pi\in\Gamma(\mu,\nu)}\sup_{(x,y)\in\operatorname{supp}(\pi)}d_{\mathbb{R}^{2},\infty}(x,y)\,.$

An important result testifying of why persistent homology is interesting is
that it is a stable construct in the following sense.

######  Theorem 2.14 (Stability theorem, [12]).

Let $(X,d)$ be a compact, triangulable metric space and let $f,g\in
C^{0}(X,\mathbb{R})$, then if we see $\operatorname{Dgm}(f)$ and
$\operatorname{Dgm}(g)$ as persistence measures, then

$d_{\infty}(\operatorname{Dgm}(f),\operatorname{Dgm}(g))\leq\left\lVert
f-g\right\rVert_{\infty}\,.$

Fixing a smaller functional space, it is possible to prove a stability theorem
for $d_{p}$ as well.

######  Theorem 2.15 (Wasserstein $p$ stability, [42]).

Let $(X,d)$ be a compact, triangulable metric space of dimension $D$ and let
$\operatorname{Lip}(\Lambda,X)$ denote the space of real valued
$\Lambda$-Lipschitz functions on $X$. Then if
$f,g\in\operatorname{Lip}(\Lambda,X)$, for every $n\geq D$

$d_{p}(\operatorname{Dgm}_{k}(f),\operatorname{Dgm}_{k}(g))\leq
C_{X,\Lambda,p}\,\left\lVert f-g\right\rVert_{\infty}^{1-\frac{n}{p}}\,,$

where $\operatorname{Dgm}_{k}(f)$ denotes the persistence measure associated
to the persistent homology in degree $k$ of $f$, $H_{k}(X,f)$.

###  2.4 Trees and barcodes

There is a correspondence between trees and barcodes described in full detail
in [35]. Starting from $T_{f}$, we can look at the longest branch (starting
from the root) of $T_{f}$. This branch corresponds to the longest bar of
$\mathcal{B}(f)$ since branches of $T_{f}$ correspond to connected components
of the superlevel sets of $f$. Next, we erase this longest branch and, on the
remaining (rooted) forest, look for the next longest branch. This will be the
second longest bar of the barcode. Proceeding iteratively in this way, we
retrieve $\mathcal{B}(f)$. An illustration of this algorithm can be found in
figure 6.

We can interpret $N^{\varepsilon}$ geometrically as being equal to the number
of leaves of $T_{f}^{\varepsilon}$. In terms of the barcode, the same
$N^{\varepsilon}$ counts the number of bars of length $\geq\varepsilon$ with
the caveat that we count the infinite bar as having length equal to the range
of $f$. As we will see, reasoning in terms of trees has some major advantages,
so in what will follow we will adopt the following convention

######  Convention 2.16.

The length of the infinite bar of $\mathcal{B}(f)$ will be set to $\sup f-\inf
f$.

Figure 6: A depiction of the first steps of the algorithm which assigns a
barcode $\mathcal{B}(f)$ to a tree $T_{f}$.

###  2.5 Integration on trees and the duality between $N^{\varepsilon}$ and
$\operatorname{Pers}_{p}^{p}$

Let us recall the following simple remark made in [35]. On a tree $T_{f}$, we
can define a notion of integration by defining the unique atomless Borel
measure $\lambda$ which is characterized by the property that every geodesic
segment on $T_{f}$ has measure equal to its length. Formally, we can express
$\lambda$ in two ways [37]

$\lambda=\int_{\mathbb{R}}dx\sum_{\begin{subarray}{c}\tau\in T_{f}\\\
f(\tau)=x\end{subarray}}\delta_{\tau}\quad\text{and}\quad\lambda=\int_{0}^{\infty}d\varepsilon\sum_{\begin{subarray}{c}\tau\in
T_{f}\\\ h(\tau)=\varepsilon\end{subarray}}\delta_{\tau}$

By using the second way of writing $\lambda$, the identity

$\lambda(T_{f}^{\varepsilon})=\int_{\varepsilon}^{\infty}N^{a}\;da$

is clear, as every sum in the second expression is finite for all
$\varepsilon>0$ and has $N^{\varepsilon}$ terms. Of course, we could very well
have written it using the first sum, but this poses the difficulty that if
$T_{f}$ is infinite, so is the sum considered in this formal expression for at
least some value of $x$. However, the restricted sum

$\sum_{\begin{subarray}{c}\tau\in T_{f}\\\ f(\tau)=x\\\
h(\tau)\geq\varepsilon\end{subarray}}\delta_{\tau}$

is finite for all $\varepsilon>0$ and there are exactly $N^{x,x+\varepsilon}$
terms in this sum. We thus obtain an alternative expression for
$\lambda(T_{f}^{\varepsilon})$

$\lambda(T_{f}^{\varepsilon})=\int_{\mathbb{R}}N^{x,x+\varepsilon}\;dx\,.$

We deduce that more information is contained in $N^{x,x+\varepsilon}$ than in
$\lambda(T_{f}^{\varepsilon})$ (and by extension than in $N^{\varepsilon}$).
The calculation above provides the connection between Chazal and Divol and
Baryshnikov’s functional $N^{x,x+\varepsilon}$ and the functional detailed in
this paper $N^{\varepsilon}$, since $N^{\varepsilon}$ is nothing other than
the derivative of $\lambda(T_{f}^{\varepsilon})$.

The study of $N^{\varepsilon}$ is in fact completely equivalent to the study
of $\operatorname{Pers}_{p}^{p}(f)$. Indeed,

$\operatorname{Pers}_{p}^{p}(f)=p\int_{T_{f}}h(\tau)^{p-1}\;\lambda(d\tau)=p\int_{0}^{\infty}\varepsilon^{p-1}N^{\varepsilon}\;d\varepsilon\,,$

where $h:T_{f}\to\mathbb{R}$ associating to $\tau\in T_{f}$ the distance
between $\tau$ and the highest leaf (with respect to the filtration of $f$)
above $\tau$ in $T_{f}$. We immediately recognize the above integral as being
the Mellin transform of $N^{\varepsilon}$. Allowing for complex $p$, this
integral relation can be inverted by virtue of the Mellin inversion theorem,
provided that the fundamental strip of $N^{\varepsilon}$ is not empty. For
compact intervals and continuous functions $f$, this fundamental strip is
never empty (provided ${\mathcal{L}}(f)<\infty$) and in fact is exactly equal
to $\langle{\mathcal{L}}(f),\infty\rangle$. Thus, for any real number
$c>{\mathcal{L}}(f)$,

$N^{\varepsilon}=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}\operatorname{Pers}^{p}_{p}(f)\,\varepsilon^{-p}\;\frac{dp}{p}\;,$

which estabilished the duality relation desired. Notice also that
$\operatorname{Pers}_{p}^{p}$ is a norm in the sense that

$\operatorname{Pers}_{p}^{p}(f)=p\left\lVert
h\right\rVert_{L^{p-1}(\lambda)}^{p-1}\,,$

For any (deterministic) continuous function $f$,
$\operatorname{Pers}_{p}^{p}(f)$ is nothing other than a sum of the bars of
the barcode to the power $p$. An in depth explanation of this is provided in
[35, §2.2], but let us briefly give some intuition for this. By the algorithm
depicted in figure 6, if we denote $b$ any of the bars of the barcode, seen as
embedded in the tree $T_{f}$ the length of the branch, $\ell(b)$, can be
written as

$p\int_{b}h(\tau)^{p-1}\lambda(d\tau)=\ell(b)^{p}\,.$

The bars of the barcode partition the tree $T_{f}$, so that the integration
present in the definition of $\operatorname{Pers}^{p}_{p}$ is nothing other
than the sum of the $\ell(b)^{p}$’s.

######  Remark 2.17.

This definition of $\operatorname{Pers}_{p}^{p}$ coincides perfectly with a
definition of $\operatorname{Pers}_{p}^{p}$ typically used in persistent
homology [15, 31, 43, 17, 10], as long as we consider that the infinite bar
has the length of the range (i.e. the $\sup-\inf$) of the function $f$. Of
course, within this framework an equally valid definition for
$\operatorname{Pers}_{p}^{p}$ would have been to exclude the infinite bar from
being counted all-together, and to consider only the bars of finite length.
This approach turns out to give the correct definition for the
$\operatorname{Pers}_{p}^{p}$-functional in the definition of tail
$\zeta$-functions (cf. definition 3.24), which is necessary to study Lévy
$\alpha$-stable processes for $\alpha<2$.

Additionally, by the usual inequalities of $L^{p}$-spaces,

######  Proposition 2.18.

$\operatorname{Pers}_{p}^{p}$ is almost $\log$-convex, i.e. let $p_{0}<p_{1}$
and $\theta\in[0,1]$ and set $p=(1-\theta)p_{0}+\theta p_{1}$, then,

$\operatorname{Pers}_{p}^{p}\leq\frac{p}{p_{0}^{1-\theta}p_{1}^{\theta}}\;\operatorname{Pers}_{p_{0}}^{p_{0}(1-\theta)}\,\operatorname{Pers}_{p_{1}}^{p_{1}\theta}\,.$

###### Proof.

The statement follows directly from an application of Lyapunov’s inequality
for $L^{p}$-spaces. ∎

More generally, it is always true that one can express the $L^{p}(\mu)$-norm
of a function $f$ as the Mellin transform of the repartition function of
$\left\lvert f\right\rvert$, $\mu(\left\lvert f\right\rvert>x)$.

####  2.5.1 Calculation of $N^{\varepsilon}$ in dimension one

In dimension one, it is possible to use the total order of $\mathbb{R}$ and
count $N^{\varepsilon}$ by counting the number of times we go up by at least
$\varepsilon$ from a local minimum and down by at least $\varepsilon$ from a
local maximum. This idea can be formalized by the following sequence,
originally introduced by Neveu et al. [32].

######  Definition 2.19.

Setting $S_{0}^{\varepsilon}=T_{0}^{\varepsilon}=0$, we define a sequence of
times recursively

$\displaystyle T_{i+1}^{\varepsilon}$ $\displaystyle:=\inf\left\\{\,t\geq
S_{i}^{\varepsilon}\;\Bigg{|}\;\sup_{[S_{i}^{\varepsilon},t]}f-f(t)>\varepsilon\right\\}$
$\displaystyle S_{i+1}^{\varepsilon}$ $\displaystyle:=\inf\left\\{\,t\geq
T_{i+1}^{\varepsilon}\;\Bigg{|}\;f(t)-\inf_{[T_{i+1}^{\varepsilon},t]}f>\varepsilon\right\\}$

Figure 7: A function $f$ in blue along with the times $T_{i}^{\varepsilon}$
and $S_{i}^{\varepsilon}$ indicated. Because of the boundary this function has
exactly 3 bars of length $\geq\varepsilon$ and not just 2.

Counting the number of bars of length $\varepsilon$ is thus exactly to count
the number of up and downs we make. More precisely

$N^{\varepsilon}=\inf\\{i\,|\,T_{i}^{\varepsilon}\text{ or
}S_{i}^{\varepsilon}=\inf\emptyset\\}$ (2.1)

by which we mean that it is the smallest $i$ such that the set over which
$T_{i}^{\varepsilon}$ or $S_{i}^{\varepsilon}$ are defined as infima is empty.

######  Notation 2.20.

We denote the range of $X$ $R$. Symbolically,

$R_{t}:=\sup_{[0,t]}X-\inf_{[0,t]}X\,.$

Moreoever, denote $N^{\varepsilon}_{t}$ the number $N^{\varepsilon}$ of the
process $X$ restricted to the interval $[0,t]$.

Intuitively, this calculation process hints at the fact that if $\varepsilon$
is small, the number of bars $N^{\varepsilon}$ should strongly depend on the
regularity of the process, as ultimately $N^{\varepsilon}$ counts the number
of “oscillations” of size $\varepsilon$. In a very precise sense, regularity
almost fully determines the asymptotics of $N^{\varepsilon}$ in the
$\varepsilon\to 0$ regime. This intuition is corroborated by the following
theorem.

######  Theorem 2.21 (Picard, §3[37] and [35]).

Given a continuous function $f:[0,1]\to\mathbb{R}$,

$\mathcal{V}(f)={\mathcal{L}}(f)=\operatorname{\overline{dim}}T_{f}=\limsup_{\varepsilon\to
0}\frac{\log N^{\varepsilon}}{\log(1/\varepsilon)}\vee 1$

where $\operatorname{\overline{dim}}$ denotes the upper-box dimension, $a\vee
b=\max\\{a,b\\}$,

$\mathcal{V}(f):=\inf\\{p\,|\,\left\lVert
f\right\rVert_{p-\text{var}}<\infty\\}\quad\text{and}\quad{\mathcal{L}}(f):=\inf\\{p\,|\,\operatorname{Pers}_{p}(f)<\infty\\}\,.$

For the rest of this paper it is exactly the functional
$\operatorname{Pers}_{p}^{p}$ which shall occupy us.

######  Remark 2.22.

The $\operatorname{Pers}_{\infty}$ functional is stable under $L^{\infty}$
perturbations of $f$. However, it is unknown whether a similar stability
result exists for $p<\infty$.

###  2.6 $\zeta$-functions associated to stochastic processes

######  Definition 2.23.

Let $f$ be a stochastic process on some compact topological space $X$. Its
$\zeta$-function $\zeta_{f}$ is defined by:

$\zeta_{f}(p):=\mathbb{E}\\!\left[\operatorname{Pers}_{p}^{p}(f)\right]=p\int_{0}^{\infty}\varepsilon^{p-1}\mathbb{E}\\!\left[N^{\varepsilon}\right]\;d\varepsilon\,.$

for $p\in\langle{\mathcal{L}}(f),\infty\rangle$.

This is reminiscent of the structure of the $\zeta$-function, but a priori not
enough to draw any parallels. However, it turns out that this nomenclature
turns out to have a meaning for stochastic processes. Similarly, we could
consider the $\operatorname{Pers}_{p}^{p}$ functional of $T_{f}^{>x}$, which
denotes the forest

$T_{f}^{>x}=\\{\tau\in T_{f}\,|\,f(\tau)>x\\}$

######  Remark 2.24.

In the tree setting the number $N^{x,x+\varepsilon}$ is also the number of
branches of length $\geq\varepsilon$ in the forest $T_{f}^{>x}$.

Following this analogy, it is natural to define

######  Definition 2.25.

The local $\zeta$-function associated to $f$ at $x$, $\zeta_{f}^{x}$ is
defined as

$\zeta^{x}_{f}(p):=\mathbb{E}\\!\left[\operatorname{Pers}_{p}^{p}(T_{f}^{>x})\right]=p\int_{0}^{\infty}\varepsilon^{p-1}\mathbb{E}\\!\left[N^{x,x+\varepsilon}\right]\;d\varepsilon\,.$

######  Proposition 2.26.

The two $\zeta$-functions we have so far defined are related via the following
formula

$\zeta_{f}(p)=p\int_{\mathbb{R}}\zeta^{x}_{f}(p-1)\;dx\,.$

###### Proof.

Let us start by noticing that $N^{\varepsilon}$ is nothing other than

$N^{\varepsilon}=-\frac{\partial}{\partial\varepsilon}\lambda(T_{f}^{\varepsilon})=-\frac{\partial}{\partial\varepsilon}\int_{\mathbb{R}}N^{x,x+\varepsilon}\;dx\,,$

where this derivative is defined and locally constant almost everywhere. From
this and the fact that the derivative $\frac{\partial
N^{x,x+\varepsilon}}{\partial\varepsilon}$ is also defined and locally
constant almost everywhere and

$N^{\varepsilon}=-\int_{\mathbb{R}}\frac{\partial
N^{x,x+\varepsilon}}{\partial\varepsilon}\;dx\,.$

Applying the Mellin transform to both sides and applying Tonelli’s theorem,

$\int_{0}^{\infty}\varepsilon^{p-1}N^{\varepsilon}\;d\varepsilon=-\int_{\mathbb{R}}dx\int_{0}^{\infty}\varepsilon^{p-1}N^{x,x+\varepsilon}\;d\varepsilon\,.$

From the derivation functional property of the Mellin transform (cf. table 1)
we get

$\int_{0}^{\infty}\varepsilon^{p-1}N^{\varepsilon}\;d\varepsilon=\int_{\mathbb{R}}dx\;(p-1)\int_{0}^{\infty}\varepsilon^{p-2}N^{x,x+\varepsilon}\;d\varepsilon\,.$

Applying the expectation to both sides, multiplying times $p$ and applying
Tonelli’s theorem once again, we have the desired result, namely,

$\zeta_{f}(p)=p\int_{\mathbb{R}}\zeta^{x}_{f}(p-1)\;dx\,.$

∎

######  Remark 2.27.

If the process starts at $0$, it is in general easy to compute $\zeta^{x}_{f}$
for $x>0$, but more challenging to do so for $x<0$. In what will follow, we
will always focus on $x>0$.

######  Notation 2.28.

For the rest of this paper we will take the following conventions. First, we
will sometimes omit the subscript $t$ of $N^{\varepsilon}_{t}$ whenever
convenient. The Laplace transform ${\mathcal{L}}$ is always taken with respect
to the variable $t$ and its conjugate variable will always be $\lambda$.
Similarly, Mellin transforms will always be taken with respect to the variable
$\varepsilon$ and its conjugate variable will be $p$.

######  Lemma 2.29.

Let $f(x,t):[0,\infty[^{2}\to\mathbb{R}_{+}$ such that the functions $f(x,-)$
and $f(-,t)$ are monotone in their arguments. Then, denoting
${\mathcal{L}}_{t}$ the Laplace transform with respect to $t$,

${\mathcal{M}}_{x}{\mathcal{L}}_{t}[f]={\mathcal{L}}_{t}{\mathcal{M}}_{x}[f]$

###### Proof.

The monotonicity of $f$ in its arguments ensures that $f$ is a measurable,
positive function. The statement holds by virtue of Tonnelli’s theorem. ∎

######  Remark 2.30.

Notice this last lemma is applicable to $N^{\varepsilon}_{t}$,
$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]$,
$\mathbb{P}(N^{\varepsilon}_{t}\geq k)$ and other such quantities.

###  2.7 Average diagrams and characterizations of Wasserstein convergence of
diagrams

######  Notation 2.31.

For the rest of this section, denote

$\mathcal{X}:=\\{(x,y)\in\mathbb{R}^{2}\,|\,y>x\\}\text{ and
}\Delta:=\\{(x,x)\in\mathbb{R}^{2}\\}\,,$

which we equip with the metric $\ell^{\infty}$-metric on $\mathbb{R}^{2}$
recalled in equation 2.3.3. Let $\varepsilon>0$ and denote
$\Delta_{\varepsilon}\subset\overline{\mathcal{X}}$ an open tubular
neighbourhood of radius $\varepsilon$ around $\Delta$ inside
$\overline{\mathcal{X}}$ and denote its complement in $\mathcal{X}$ by
$\Delta_{\varepsilon}^{c}$. Finally, denote

$R_{x,\varepsilon}:=\,]\\!-\\!\infty,x]\times[x+\varepsilon,\infty[\,.$

By looking at persistence diagrams of functions as measures (cf. section 2.3.3
for details), it is possible to define a notion of an average diagram of a
stochastic process.

######  Definition 2.32.

Let $M$ be a compact, triangulable metric space, $\mathcal{E}(M)$ be a Polish
metric space of (q-tame) functions on $M$ and let $\mathcal{D}$ denote the
space of measures on the upper half-plane $\mathcal{X}$. Let
$\phi:\mathcal{E}(M)\to\mathcal{D}$ be the map taking
$f\mapsto\operatorname{Dgm}(f)$, where $\operatorname{Dgm}(f)$ is seen as a
measure, and let $X:\Omega\to\mathcal{E}(M)$ be a stochastic process of law
$\mu$ on $\mathcal{E}(M)$. Then, the average diagram of $X$ is given by

$\mathbb{E}\\!\left[\operatorname{Dgm}(X)\right]:=\mathbb{E}_{\mu}[\phi]=\int_{\mathcal{E}(M)}\phi(f)\;d\mu(f)\,.$

######  Remark 2.33.

Note that $\mathbb{E}\\!\left[\operatorname{Dgm}(X)\right]$ is itself a
measure. Whenever this measure is absolutely continuous with respect to the
Lebesgue measure on $\mathcal{X}$, there exists $g:\mathcal{X}\to\mathbb{R}$
such that for any test function $f:\mathcal{X}\to\mathbb{R}$

$\mathbb{E}\\!\left[\operatorname{Dgm}(X)\right](f)=\int_{\mathcal{X}}g(x,y)f(x,y)\;dx\,dy\,.$

A sufficient condition to ensure the existence of $g$ is that
$\partial_{x}\partial_{y}\mathbb{E}\\!\left[N^{x,y}\right]$ exists (this is
equivalent to requiring the existence of
$\partial_{x}\partial_{\varepsilon}\mathbb{E}\\!\left[N^{x,x+\varepsilon}\right]$).
Instead of using birth-death coordinates, we may also express this density in
terms of birth-persistence coordinates. In these coordinates, we will denote
this density function by $g(x,\varepsilon)$, somewhat abusing the notation.

######  Definition 2.34.

The space of measures on $\overline{\mathcal{X}}$ with finite
$\operatorname{Pers}_{p}$ will be denoted $\mathcal{D}_{p}$. Symbolically,

$\mathcal{D}_{p}:=\left\\{\mu\in\mathcal{D}\,\big{|}\,\operatorname{Pers}_{p}(\mu):=d_{p}(\mu,\Delta)<\infty\right\\}\,,$

where $d_{p}$ is the Wasserstein $p$-distance on the space of diagrams defined
in [17] and explicited in definition 2.13.

This allows us to define a $\zeta$-function for $\mu\in\mathcal{D}_{p}$ as
follows.

######  Definition 2.35.

Let $\mu\in\mathcal{D}_{p}$ and suppose that for some $q>p$

$\mu(\Delta_{\varepsilon}^{c})=O(\varepsilon^{-q})\quad\text{as
}\varepsilon\to\infty\,,$

then define the $\zeta$-function associated to $\mu$ by

$\zeta_{\mu}(p):=\operatorname{Pers}_{p}^{p}(\mu)=p\,{\mathcal{M}}\\!\left[\mu(\Delta_{\varepsilon}^{c})\right](p)\,.$

######  Remark 2.36.

A priori, $\zeta_{\mu}(p)$ is defined on the strip $\langle
p,q\rangle\subset\mathbb{C}$. This could have also been guaranteed by
replacing the condition of decay of $\mu(\Delta_{\varepsilon}^{c})$ by
requiring that $\mu\in\mathcal{D}_{p}\cap\mathcal{D}_{q}$.

As before, we may also define a local $\zeta$-function.

######  Definition 2.37.

The local $\zeta$-function associated to $\mu$ at $x$ is defined as

$\zeta^{x}_{\mu}(p):=p\,{\mathcal{M}}[\mu(R_{x,\varepsilon})](p)\,.$

######  Remark 2.38.

These definitions are compatible with the notions of $\zeta$-functions defined
for a stochastic process. Seeing
$\mathbb{E}\\!\left[\operatorname{Dgm}(X)\right]$ as a measure on
$\overline{\mathcal{X}}$,
$\zeta_{X}=\zeta_{\mathbb{E}\\!\left[\operatorname{Dgm}(X)\right]}$ (and the
same holds for local $\zeta$-functions).

A characterization of the topology metrized by the distance $d_{p}$ is useful
and has been investigated by Divol and Lacombe in [17]. The reader familiar
with optimal transport will recognize this as an adaptation of the known
characterization for probability measures of Wasserstein topology by vague
convergence and convergence of $p$th-moments. That this equivalence holds for
measures of a priori infinite mass is, however, a non-trivial extension.

######  Lemma 2.39 (Characterization of the topology metrized by $d_{p}$,
[17]).

Let $(\mu_{n})_{n}\subset\mathcal{D}_{p}$ and $\mu\in\mathcal{D}_{p}$. Then,
the following equivalence holds

$\left\\{d_{p}(\mu_{n},\mu)\xrightarrow{n\to\infty}0\right\\}\iff\left\\{\mu_{n}\xrightarrow[n\to\infty]{v}\mu\text{
and
}\operatorname{Pers}_{p}(\mu_{n})\xrightarrow[n\to\infty]{}\operatorname{Pers}_{p}(\mu)\right\\}\,.$

######  Remark 2.40.

Of course, given our choice of notation, if $p<\infty$, we can rewrite
$\operatorname{Pers}_{p}(\mu_{n})\to\operatorname{Pers}_{p}(\mu)$ as
$\zeta_{\mu_{n}}(p)\to\zeta_{\mu}(p)$.

Furthermore, it is possible to show that

######  Proposition 2.41 (Interpolation for optimal transport).

Let $1\leq p<q\leq\infty$ and $\theta\in\,]0,1[$. Define $p_{\theta}$ by

$\frac{1}{p_{\theta}}=\frac{\theta}{p}+\frac{1-\theta}{q}\,.$

Then, for $\mu,\nu\in\mathcal{D}_{p}\cap\mathcal{D}_{q}$

$d_{p_{\theta}}(\mu,\nu)\leq
2^{1-\theta}\;d_{p}^{\theta}(\mu,\nu)\,(\operatorname{Pers}_{q}(\mu)+\operatorname{Pers}_{q}(\nu))^{1-\theta}\,.$

Consequently, if $p\leq r\leq q$, then
$\mathcal{D}_{r}\subset\mathcal{D}_{p}\cap\mathcal{D}_{q}$.

######  Remark 2.42.

For probability measures, Wasserstein interpolation follows trivially from an
application of Jensen’s inequality. However, since diagrams are a priori of
infinite mass, this interpolation result needs to be shown.

###### Proof.

Let $\pi$ be an optimal transport for $d_{p}$. Applying Littlewood’s
inequality,

$\displaystyle d_{p_{\theta}}(\mu,\nu)$ $\displaystyle\leq\left\lVert
d_{\mathbb{R}^{2},\infty}\right\rVert_{L^{p_{\theta}}(\pi)}\leq\left\lVert
d_{\mathbb{R}^{2},\infty}\right\rVert_{L^{p}(\pi)}^{\theta}\left\lVert
d_{\mathbb{R}^{2},\infty}\right\rVert_{L^{q}(\pi)}^{1-\theta}$
$\displaystyle=d_{p}(\mu,\nu)^{\theta}\left[\int_{\overline{\mathcal{X}}^{2}}d_{\mathbb{R}^{2},\infty}^{q}(z,z^{\prime})\;d\pi(z,z^{\prime})\right]^{\frac{1-\theta}{q}}$
$\displaystyle\leq
d_{p}(\mu,\nu)^{\theta}\left[2^{q}\int_{\overline{\mathcal{X}}^{2}}d_{\mathbb{R}^{2},\infty}^{q}(z,\Delta)+d_{\mathbb{R}^{2},\infty}^{q}(\Delta,z^{\prime})\;d\pi(z,z^{\prime})\right]^{\frac{1-\theta}{q}}\,,$

where this equality holds everywhere on the support of $\pi$. This can be
shown by defining

$S=\\{(z,z^{\prime})\in\overline{\mathcal{X}}^{2}\cap\operatorname{supp}(\pi)\,|\,d_{\mathbb{R}^{2},\infty}(z,z^{\prime})>d_{\mathbb{R}^{2},\infty}(z,\Delta)+d_{\mathbb{R}^{2},\infty}(z^{\prime},\Delta)\\}\,.$

This set $S$ either has null or positive measure. If it has positive measure,
then we can modify the transport plan $\pi$ by sending the projections of $S$
to the diagonal, thereby producing a transport plan of strictly inferior cost
to that of $\pi$, which is a contradiction. Hence, $S$ is of null measure, so
the equality holds over the support of the measure. Finally, this entails

$\displaystyle
d_{p_{\theta}}(\mu,\nu)=2^{1-\theta}\;d_{p}(\mu,\nu)^{\theta}\,(\operatorname{Pers}_{q}(\mu)+\operatorname{Pers}_{q}(\nu))^{1-\theta}\,.$

If $q=\infty$, since $\pi$ is an optimal transport between $\mu$ and $\nu$ and
$\mu,\nu\in\mathcal{D}_{\infty}$, $\pi$ itself must have compact support and
the diameter of the support is bounded above by
$\operatorname{Pers}_{\infty}(\mu)\vee\operatorname{Pers}_{\infty}(\nu)$, so
the inequality of the proposition follows. ∎

######  Lemma 2.43 (Sequential continuity of the inverse Mellin transform).

Let $f_{k}:\;]0,\infty[\to\mathbb{C}$ be a sequence of functions uniformly
bounded by a function $g:\,]0,\infty[\to\mathbb{R}_{+}$, whose Mellin
transform $g^{*}$ is defined over some non-empty strip
$\langle\alpha,\beta\rangle\subset\mathbb{C}$. Suppose further that there is a
function $f:\;]0,\infty[\to\mathbb{C}$ such that
$f^{*}:\langle\alpha,\beta\rangle\to\mathbb{R}_{+}$ and $f_{k}^{*}\to f^{*}$
uniformly on every compact set of $\langle\alpha,\beta\rangle$. Then,
$f_{k}\to f$ almost everywhere.

###### Proof.

For every $k$, $\left\lvert f_{k}\right\rvert\leq g\vee\left\lvert
f\right\rvert$ and so the sequence $(f_{k})_{k}$ is bounded in a weighted
$L^{1}$ space. Up to extraction of a subsequence, $f_{k}$ converges a.e. to
some function $h$, also bounded above by $g\vee\left\lvert f\right\rvert$. By
dominated convergence, along this subsequence, $f_{k}^{*}\to h^{*}$, which
entails that $h=f$ a.e. since the Mellin transform is injective and
$(h-f)^{*}=0$ identically on $\langle\alpha,\beta\rangle$, since along any
subsequence $f_{k}^{*}\to f^{*}$ uniformly on every compact set of
$\langle\alpha,\beta\rangle$. It follows that the sequence $(f_{k})_{k}$ has
$f$ as its only accumulation point, finishing the proof. ∎

######  Remark 2.44.

It is sufficient to consider that for all $s\in\,]\alpha,\beta[$, the sequence
$(x^{s-1}f_{k}(x))_{k}$ is absolutely uniformly integrable.

Both of these lemmas allow for a comprehensive characterization of the objects
we have thus far been concerned with throughout this paper.

######  Theorem 2.45 ($\zeta$ characterization of Wasserstein
$p$-convergence).

Let $(\mu_{n})_{n}\subset\mathcal{D}_{p}\cap\mathcal{D}_{q}$ be a sequence of
q-tame measures and $\mu\in\mathcal{D}_{p}\cap\mathcal{D}_{q}$ and suppose
that the sequence $(\mu_{n}(\Delta_{\varepsilon}^{c}))_{n}$ can be uniformly
bounded above by a function $g:\;]0,\infty[\,\to\mathbb{R}_{+}$ such that for
$\varepsilon\in\;]0,1]$ $g(\varepsilon)=O(\varepsilon^{-p})$ as
$\varepsilon\to 0$ and on $[1,\infty[$, $g(\varepsilon)=O(\varepsilon^{-q})$
as $\varepsilon\to\infty$. Then, the following are equivalent:

  1. 1.

There exists $p<r<q$ such that $d_{r}(\mu_{n},\mu)\xrightarrow{n\to\infty}0$.

  2. 2.

There exists $p<r<q$ such that $\mu_{n}\xrightarrow[n\to\infty]{v}\mu$ and
$\zeta_{\mu_{n}}(r)\to\zeta_{\mu}(r)$.

  3. 3.

For almost every $x\in\mathbb{R}$ and $\varepsilon>0$,
$\mu_{n}(\Delta_{\varepsilon}^{c})\to\mu(\Delta_{\varepsilon}^{c})$ and
$\mu_{n}(R_{x,\varepsilon})\to\mu(R_{x,\varepsilon})$.

  4. 4.

For almost every $x\in\mathbb{R}$, $\zeta^{x}_{\mu_{n}}\to\zeta^{x}_{\mu}$ and
$\zeta_{\mu_{n}}\to\zeta_{\mu}$ uniformly on every compact of $\langle
p,q\rangle$.

  5. 5.

For all $p<r<q$, $d_{r}(\mu_{n},\mu)\xrightarrow{n\to\infty}0$.

###### Proof.

Let us break down the proof in different steps.

  * •

$(1)\iff(2)$ by lemma 2.39.

  * •

$(2)\implies(3)$. By the Portmanteau theorem, the vague convergence of
$\mu_{n}$ to $\mu$ entails that for any continuity set $A$ of $\mu$,
$\mu_{n}(A)\to\mu(A)$. In particular, since the $\mu_{n}$ and $\mu$ are
q-tame, $R_{x,\varepsilon}$ is a continuity set of $\mu$ for almost every $x$
and $\varepsilon$, so $\mu_{n}(R_{x,\varepsilon})\to\mu(R_{x,\varepsilon})$
and $\mu_{n}(\Delta_{\varepsilon}^{c})\to\mu(\Delta_{\varepsilon}^{c})$ a.e..

  * •

$(3)\implies(4)$. Since
$\mu_{n}(\Delta_{\varepsilon}^{c})\to\mu(\Delta_{\varepsilon}^{c})$ a.e., if
we let $s\in\langle\alpha,\beta\rangle$, then

$\displaystyle\left\lvert\zeta_{\mu_{n}}(s)-\zeta_{\mu}(s)\right\rvert$
$\displaystyle\leq\left\lvert
s\right\rvert\int_{0}^{\infty}\varepsilon^{\operatorname{Re}(s)-1}\left\lvert\mu_{n}(\Delta_{\varepsilon}^{c})-\mu(\Delta_{\varepsilon}^{c})\right\rvert\;d\varepsilon$
$\displaystyle=\left\lvert
s\right\rvert\left\\{\int_{0}^{1}+\int_{1}^{\infty}\right\\}\varepsilon^{\operatorname{Re}(s)-1}\left\lvert\mu_{n}(\Delta_{\varepsilon}^{c})-\mu(\Delta_{\varepsilon}^{c})\right\rvert\;d\varepsilon$

The domination conditions of the theorem on
$\mu_{n}(\Delta_{\varepsilon}^{c})$ and $\mu(\Delta_{\varepsilon}^{c})$
guarantee that this quantity is integrable as soon as
$p<\operatorname{Re}(s)<q$. By dominated convergence,
$\zeta_{\mu_{n}}\to\zeta_{\mu}$ on every compact of $\langle p,q\rangle$. We
apply the same reasoning to the $R_{x,\varepsilon}$ by noting that the measure
of $R_{x,\varepsilon}$ is always dominated by that of
$\Delta_{\varepsilon}^{c}$.

  * •

$(4)\implies(5)$. Fix $r\in\;]p,q[$ and take a compact set $K$ around $r$
contained in $\langle p,q\rangle$. It suffices thus to show that the
convergence of these $\zeta$-functions entails vague convergence of the
$\mu_{n}$, which will show the result by lemma 2.39. Denoting
$A_{\varepsilon}\in\\{R_{x,\varepsilon},\Delta_{\varepsilon}^{c}\\}$, the
boundedness condition of the theorem on $\mu_{n}(A_{\varepsilon})$ entails
that, by lemma 2.43, $\mu_{n}(A_{\varepsilon})\to\mu(A_{\varepsilon})$ almost
everywhere. Since the collection of $R_{x,\varepsilon}$ and
$\Delta_{\varepsilon}^{c}$ together forms a $\pi$-system, so for every
continuity set of $\mu$, we have convergence of their measures $\mu_{n}$ to
$\mu$, and so by applying the Portmanteau theorem once again, $\mu_{n}\to\mu$
vaguely, which shows the result.

  * •

$(5)\implies(1)$ trivially.

∎

######  Remark 2.46.

Of course, by interpolation, it is sufficient that for all $\mu$ and
$\operatorname{Re}(s)>p$, $\zeta_{\mu_{n}}(s)$ be bounded to guarantee that
for any $s\in\;]p,\infty[$, $d_{s}(\mu_{n},\mu)\to 0$, if
$d_{p}(\mu_{n},\mu)\to 0$.

######  Remark 2.47.

Theorem 2.45 applies to average diagrams from any stochastic process
satisfying the hypotheses of the theorem. This entails that if the
$\zeta$-functions of a process converge on some open set
$\langle\alpha,\beta\rangle\subset\mathbb{C}$, the underlying expected
diagrams themselves converge in some Wasserstein distance.

##  3 $\zeta$-functions of Lévy processes and semimartingales

###  3.1 Semimartingales

A first important motivating result regarding $\zeta$-functions is the
following.

######  Proposition 3.1.

Let $X$ be a continuous semimartingale $X=M+A$ on the interval $[0,t]$ such
that for $s\geq 1$

$\mathbb{E}\\!\left[[M]_{t}^{s/2}+\left(\int_{0}^{t}\left\lvert
dA\right\rvert_{s}\right)^{s}\,\right]<\infty\,.$

Then, the (local) $\zeta$-function of $X$ is meromorphic on
$\operatorname{Re}(p)\geq 2$ (resp. $\operatorname{Re}(p)\geq 1$) with a
single simple pole at $p=2$ (resp. $p=1$). Furthermore, if $X$ is a continuous
semimartingale (and some conditions) and $[X]_{t}<\infty$, then in $L^{s}$

$N^{\varepsilon}_{X}\sim\frac{[X]_{t}}{2\varepsilon^{2}}\quad\text{as
}\varepsilon\to 0\,.$

######  Remark 3.2.

By quadratic variation, we mean the quadratic variation in the probabilistic
sense, i.e. in the sense that we take the limit of the variation of the
process as the mesh of the partition of the interval $[0,t]$ goes to zero, as
opposed to the real quadratic variation, where the supremum over all
partitions is considered.

###### Proof.

The statement holds by the almost sure existence of continuous modifications
of local times for continuous semimartingales and the fact that uniformly in
$x$ and $t$, it is known [41, Ch. VI Thm. 1.10] that under the technical
hypothesis of the theorem,

$2\varepsilon N^{x,x+\varepsilon}_{X}\xrightarrow[\varepsilon\to
0]{L^{s}}L^{x}_{X}(t)\,.$

Recall the density occupation formula for the local times of a continuous
semimartinglale $X$ [28, Cor. 9.7], which states that, almost surely, for
every $t\geq 0$ and any non-negative measurable function $\phi$ on
$\mathbb{R}$

$\int_{0}^{t}\phi(X_{\tau})\;d[X]_{\tau}=\int_{\mathbb{R}}\phi(a)L^{a}_{X}(t)\;da\,.$

Taking $\phi$ to be the constant function $1$, we get that in $L^{s}$

$\displaystyle\lambda(T_{f}^{\varepsilon})$
$\displaystyle=\int_{\mathbb{R}}N^{x,x+\varepsilon}\;dx\sim\frac{1}{2\varepsilon}\int_{\mathbb{R}}L_{X}^{x}(t)\;dx\;\;\text{as
}\varepsilon\to 0$
$\displaystyle=\frac{1}{2\varepsilon}\int_{0}^{t}d[X]_{\tau}=\frac{[X]_{t}}{2\varepsilon}+o(\varepsilon^{-1})\;\;\text{as
}\varepsilon\to 0\,.$

But for $\delta>0$ small enough, by monotonicity of $N^{\varepsilon}$,

$N^{(1+\delta)\varepsilon}\leq\frac{\lambda(T_{f}^{\varepsilon})-\lambda(T_{f}^{(1+\delta)\varepsilon})}{\delta\varepsilon}=\frac{1}{\delta\varepsilon}\int_{\varepsilon}^{(1+\delta)\varepsilon}N^{a}\;da\leq
N^{\varepsilon}\,,$

so that in $L^{s}$ for every $\delta>0$ small enough,

$\frac{[X]_{t}}{2\varepsilon^{2}}\frac{1}{1+\delta}\lesssim
N^{\varepsilon}\lesssim\frac{[X]_{t}}{2\varepsilon^{2}}\frac{1}{1-\delta}\quad\text{as
}\varepsilon\to 0\,.$

Since $\delta$ can be taken to be arbitrarily small

$N^{\varepsilon}\sim\frac{[X]_{t}}{2\varepsilon^{2}}\quad\text{as
}\varepsilon\to 0\,$

in $L^{s}$ as desired. ∎

######  Remark 3.3.

If the continuous semimartingale is Brownian motion, the same arguments as
well as a result regarding the representation by downcrossings of the local
time by Itô [25, §2.4] show that this asymptotic relation holds in fact almost
surely.

######  Corollary 3.4.

If $X$ is a continuous semimartingale, then in expectation
$\operatorname{Pers}_{p}^{p}(X)$ admits a pole of order $1$ at $p=2$ of
residue $[X]_{t}$.

In [36], we have already studied the functional $N^{\varepsilon}$ for Markov
processes. Let us briefly recall some useful known facts about
$N^{\varepsilon}$.

######  Proposition 3.5 (P, [36]).

Using summation by parts, it is possible to write

$\mathbb{E}\\!\left[(N^{\varepsilon}_{t})^{s}\right]=\sum_{k\geq
1}(k^{s}-(k-1)^{s})\mathbb{P}(N^{\varepsilon}_{t}\geq k)$

For processes on the interval which are not periodic (in the sense of [36]),
if $k\geq 2$

$\mathbb{P}(N_{t}^{\varepsilon}\geq k)=\mathbb{P}(S_{k-1}^{\varepsilon}\leq
t)\,,$ (3.2)

and $\mathbb{P}(N_{t}\geq 1)=\mathbb{P}(R_{t}\geq\varepsilon)$. Furthermore if
$X$ has the strong Markov property,

$\mathbb{E}\\!\left[N_{t}^{\varepsilon}\right]\sim\mathbb{P}(R_{t}\geq\varepsilon)\quad\text{as
}\varepsilon\to\infty\,.$

Finally, for $k\geq 2$ the Laplace transform (with respect to time, as per our
convention) of equation 3.2 is

${\mathcal{L}}(\mathbb{P}(N_{t}^{\varepsilon}\geq
k))(\lambda)=\frac{\mathbb{E}\\!\left[e^{-\lambda
S_{k-1}^{\varepsilon}}\right]}{\lambda}\,.$ (3.3)

###### Proof of proposition 3.5.

The only thing to prove is the result of equation 3.3, as the previous
statements are all proved in [36]. Since we are dealing with processes which
are not periodic in the sense of [36], then

$\mathbb{P}(N_{t}^{\varepsilon}\geq k)=\mathbb{P}(S_{k-1}^{\varepsilon}\leq
t)\,,$

since as soon as the hitting time $S_{k-1}^{\varepsilon}$ is attained we have
at least $k$ bars (due to the boundary of the interval). Using standard
functional properties of the Laplace transform it is easy to see that

${\mathcal{L}}[\mathbb{P}(S_{k-1}^{\varepsilon}\leq
t)](\lambda)=\frac{{\mathcal{L}}[\mathbb{P}(S_{k-1}^{\varepsilon}=t)](\lambda)}{\lambda}\,,$

where $\mathbb{P}(S_{k-1}^{\varepsilon}=t)$ denotes the probability density
function of $S_{k-1}^{\varepsilon}$. However, the Laplace transform of this
density function is nothing other than the moment generating function
$\mathbb{E}\\!\left[e^{-\lambda S_{k-1}^{\varepsilon}}\right]$, since
$S_{k-1}^{\varepsilon}$ is a positive random variable. ∎

######  Remark 3.6.

If the process has the strong Markov property, we can write
$\mathbb{E}\\!\left[e^{-\lambda S_{k-1}^{\varepsilon}}\right]$ as the product
of the Laplace transform of the distribution of its increments

$S_{k-1}^{\varepsilon}=\sum_{i=0}^{k}(S_{i}^{\varepsilon}-T_{i}^{\varepsilon})+(T_{i}^{\varepsilon}-S_{i-1}^{\varepsilon})\,.$

The expression of $\mathbb{E}\\!\left[e^{-\lambda
S_{k-1}^{\varepsilon}}\right]$ is particularly simple as soon as these
increments are independent and identically distributed.

######  Remark 3.7.

Ordering the bars of the barcode of a function $f$ by their length, and
denoting the length of the $k$th longest branch by $\ell_{k}$, the following
equivalence holds

$N_{t}^{\varepsilon}\geq k\iff\ell_{k}\geq\varepsilon\,.$

The probability distribution of both of the events above are thus the same.
Consequently, there is a one-to-one correspondence between the elements of the
sums

$\mathbb{E}\\!\left[\operatorname{Pers}^{p}_{p}\right]=\sum_{k\geq
1}\mathbb{E}\\!\left[\ell_{k}^{p}\right]=p\sum_{k\geq
1}{\mathcal{M}}[\mathbb{P}(N^{\varepsilon}_{t}\geq k)](p)$

whenever these quantities are defined. In particular, the distribution of each
bar is in principle readily available, since
$\mathbb{E}\\!\left[\ell_{k}^{p-1}\right]$ is the Mellin transform of the
distribution of $\ell_{k}$. We will later see that in particular cases, we can
gain access to the explicit distribution of bars in this way (cf. section
3.3.2).

###  3.2 Renewal theory

Throughout this section, we shall consider that the stopping times
$S_{k-1}^{\varepsilon}$ are such that the sequence
$(U_{k}^{\varepsilon})_{k\geq 0}$ is a sequence of i.i.d. atomless random
variables, where

$U_{k}^{\varepsilon}:=S_{k}^{\varepsilon}-S_{k-1}^{\varepsilon}\,.$

We shall adapt and enounce some theorems from renewal theory (cf. Allan Gut’s
book [23] for a detailed account of this) Let us define
$\hat{N}^{\varepsilon}_{t}$ by

$\hat{N}^{\varepsilon}_{t}:=\max\\{k\,|\,S_{k}\leq t\\}\,.$

Tautologically, for every $k$, $\hat{N}^{\varepsilon}_{t}\geq k\iff
S_{k}^{\varepsilon}\leq t$. In fact, adopting the convention that the infinite
bar has length the range of the process, we can write, always under convention
2.16 that

$N_{t}^{\varepsilon}=1_{\\{R_{t}\geq\varepsilon\\}}+\hat{N}^{\varepsilon}_{t}$

######  Remark 3.8.

Notice that convention 2.16 for the infinite bar might not be the most natural
or more convenient from this point of view. Indeed, if we had adopted the
convention that the infinite bar always has length at least $\varepsilon$,
then

$N_{t}^{\varepsilon}=1+\hat{N}^{\varepsilon}_{t}$

and $N^{\varepsilon}_{t}$ is a first passage time process defined by

$N^{\varepsilon}_{t}:=\min\\{k\,|\,S_{k}>t\\}\,.$

Nonetheless, as previously stated, we will keep adopting convention 2.16 for
the rest of this paper.

According to renewal theory, $\hat{N}^{\varepsilon}_{t}$ posses various
desirable limit theorems which we shall briefly recall and refer the reader to
[23] for complete proofs of the latter.

######  Theorem 3.9 (Strong Law of Counting Processes, Theorem 5.1 [23]).

Let $0<\mathbb{E}\\!\left[U_{1}^{\varepsilon}\right]<\infty$, then

$\frac{\hat{N}^{\varepsilon}_{t}}{t}\xrightarrow[a.s]{t\to\infty}\frac{1}{\mathbb{E}\\!\left[U_{1}^{\varepsilon}\right]}\quad\text{and}\quad\frac{\mathbb{E}\\!\left[(\hat{N}^{\varepsilon}_{t})^{s}\right]}{t^{s}}\xrightarrow{t\to\infty}\frac{1}{(\mathbb{E}\\!\left[U_{1}^{\varepsilon}\right])^{s}}$

for all $s>0$. If $\mathbb{E}\\!\left[U^{\varepsilon}_{1}\right]=\infty$, then
the limits are $0$.

######  Theorem 3.10 (CLT for Counting Processes, Theorem 5.2 [23]).

Let $0<\mathbb{E}\\!\left[U_{1}^{\varepsilon}\right]=\mu<\infty$ and
$\sigma^{2}=\text{Var}(U_{1}^{\varepsilon})<\infty$, then

$\frac{\hat{N}_{t}^{\varepsilon}-t/\mu}{\sqrt{\frac{\sigma^{2}t}{\mu^{3}}}}\xrightarrow[\mathbb{P}]{t\to\infty}\mathcal{N}(0,1)$

Furthermore,

$\displaystyle\mathbb{E}\\!\left[\hat{N}_{t}^{\varepsilon}\right]$
$\displaystyle=\frac{t}{\mu}+\frac{\sigma^{2}-\mu^{2}}{2\mu^{2}}+o(1)\quad\text{as
}t\to\infty$ $\displaystyle\text{Var}(\hat{N}_{t}^{\varepsilon})$
$\displaystyle=\frac{\sigma^{2}t}{\mu^{3}}+o(t)\quad\text{as }t\to\infty\,.$

These results are in particular applicable for Lévy processes, where it is
possible to show that the requirement that the sequence $(U_{k})_{k}$ is
i.i.d. is satisfied. In particular, the theorems above give the asymptotic
long-time behaviour of the number of bars.

###  3.3 Lévy processes

For Lévy processes, the small scale asymptotics of $N^{\varepsilon}$ can also
be studied up to the following caveat : a wide range of Lévy processes have
almost surely discontinuous paths (but nonetheless càdlàg), but our
construction of trees (as done in [35]) is based on continuous functions. For
this reason, it is necessary to define what tree we associate to a process $X$
when $X_{t}$ has almost surely discontinuous paths. Luckily, this caveat has
been treated for càdlàg processes in [19, 37]. We will adopt the approach
taken by Picard in [37], where the reader can find the details of the
construction. Loosely speaking, Picard’s approach consists in “completing” the
function at the discontinuity points by joining an imaginary line linking the
points of discontinuity (cf. figure 8).

Figure 8: A depiction of the construction of a tree associated to a càdlàg
function. The figure is taken from [37]

In any case, it has been shown that on some fixed interval $[0,t]$, it is
possible to obtain the behaviour of the number of bars of length
$\geq\varepsilon$ as $\varepsilon\to 0$.

######  Proposition 3.11 (Picard, §3 [37]).

Let $X$ be a Lévy process and suppose that, almost surely $X$ has no interval
on which it is monotone. Define

$\xi(\varepsilon):=\mathbb{E}\\!\left[S^{\varepsilon}+T^{\varepsilon}\right]$

for

$S^{\varepsilon}:=\inf\\{t\,|\,X_{t}-\inf_{[0,t]}X>\varepsilon\\}\quad\text{
and }T^{\varepsilon}:=\inf\\{t\,|\,\sup_{[0,t]}X-X_{t}>\varepsilon\\}\,,$

then $\xi(\varepsilon)N^{\varepsilon}\to 1$ as $\varepsilon\to 0$ in
probability. If $\xi(\varepsilon)=O(\varepsilon^{\alpha})$ for some $\alpha$,
then the convergence is almost sure.

######  Remark 3.12.

The hypothesis on $X$ is satisfied if $X$ or $-X$ is not the sum of a
subordinator and a compound Poisson process, in which case $T_{X}$ is finite,
so $N^{\varepsilon}$ is bounded. Furthermore, the convergence is always almost
sure for $\alpha$-stable processes for which $\left\lvert X\right\rvert$ is
not a subordinator by the scaling property. In fact, in that case there exists
a constant $C_{\alpha}$ such that almost surely,

$N^{\varepsilon}\sim\frac{C_{\alpha}}{\varepsilon^{\alpha}}\quad\text{as
}\varepsilon\to 0\,.$

If we can quantify correction terms to this asymptotic relation in $L^{1}$,
this gives rise to a statistical test for $\alpha$ by using the stability
results discussed in [36], we will explore this in more detail in section
3.3.3. By the self-similarity of $\alpha$-stable process following the
arguments of [37, §3], we can already at least conclude that

$\left\lvert\mathbb{E}\\!\left[N^{\varepsilon}_{\alpha}\right]-C_{\alpha}\varepsilon^{-\alpha}\right\rvert\leq
1\,.$

######  Notation 3.13.

In what will follow, we will denote by $S^{\varepsilon}$ and $T^{\varepsilon}$
two independent random variables distributed as the analogously denoted ones
in proposition 3.11. Furthermore, define
$U^{\varepsilon}=T^{\varepsilon}+S^{\varepsilon}$. In particular, if
$\varepsilon=1$, abusing the notation we will denote $U^{1}=U$.

######  Remark 3.14.

Henceforth, unless otherwise specified, we will always assume that $X$ almost
surely has no interval on which it is monotone.

This result by Picard is exactly the Strong Law of Counting Processes (theorem
3.9) applied to the stopping times we previously considered. Self-similarity
allows us to trade the $t\to\infty$ limit by an $\varepsilon\to 0$ limit.
However, Picard’s result is more general that what could’ve been deduced with
self-similarity, as it applies to Lévy processes which are not necessarily
$\alpha$-stable. In this setting, one could ask whether it is possible to
prove an analogous theorem to the CLT of Counting Processes (theorem 3.10).

######  Theorem 3.15.

Let $X$ be a Lévy process such that almost surely $X$ has no interval on which
it is monotone, using the notation defined in 3.13,

$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]=\frac{t}{\mathbb{E}\\!\left[U^{\varepsilon}\right]}+\left(\frac{\mathbb{E}\\!\left[(U^{\varepsilon})^{2}\right]}{2\mathbb{E}\\!\left[U^{\varepsilon}\right]^{2}}-1\right)+\mathbb{P}(R_{t}\geq\varepsilon)+o(\rho_{\varepsilon}^{-n})\quad\text{as
}\varepsilon\to 0\,.$

for any $n\in\mathbb{N}$, where $\rho_{\varepsilon}$ denotes the radius of
convergence of the Taylor series of $\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]$ around $\lambda=0$, which can be bounded below by
$-\log(\mathbb{P}(T^{\varepsilon}>1)\vee\mathbb{P}(S^{\varepsilon}>1))$, which
is larger than $1$ for $\varepsilon$ small enough. Furthermore, if $X$ is
$\alpha$-stable, the formula above becomes

$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]=\frac{t}{\mathbb{E}\\!\left[U\right]\varepsilon^{\alpha}}+\frac{\mathbb{E}\\!\left[U^{2}\right]}{2\mathbb{E}\\!\left[U\right]^{2}}+o(\varepsilon^{\alpha
n})\quad\text{as }\varepsilon\to 0\,.$ (3.4)

Let us stress that the main addition over the CLT of Counting Processes
(theorem 3.10) is two-fold :

  * •

We have found a similar limit for a processes which is not self-similar, and
so for which the $t\to\infty$ limit was a priori not interchangeable with the
$\varepsilon\to 0$ limit.

  * •

We have further specified the rate of decay of the remainder in terms of
$\varepsilon$, and this will turn out to be of importance;

To show the theorem, it is convenient to show first some technical lemmata,
one of which is a slight refinement to a technical lemma proved in [37].

######  Lemma 3.16 (Picard, Proposition 3.14 [37]).

The variables $S^{\varepsilon}$ and $T^{\varepsilon}$ admit finite moments of
order $k$ for all $k$ and the moment generating function
$\mathbb{E}\\!\left[e^{-\lambda U^{\varepsilon}}\right]$ is well defined on a
neighborhood of zero. Furthermore, the radius of convergence of
$\mathbb{E}\\!\left[e^{-\lambda U^{\varepsilon}}\right]$ around $0$,
$\rho_{\varepsilon}$, then

$\rho_{\varepsilon}\geq-\log(\mathbb{P}(S^{\varepsilon}>1)\vee\mathbb{P}(T^{\varepsilon}>1))\,,$

if $X$ is $\alpha$-stable, then
$\rho_{\varepsilon}=\rho\varepsilon^{-\alpha}$, for some constant $\rho>0$
(which might be infinite). Finally, there exists $1\leq C_{k}\leq
2^{k}\operatorname{Li}_{-k}(\frac{1}{2})$ such that

$\mathbb{E}\\!\left[U^{\varepsilon}\right]^{k}\leq\mathbb{E}\\!\left[(U^{\varepsilon})^{k}\right]\leq
C_{k}\,\mathbb{E}\\!\left[U^{\varepsilon}\right]^{k}\,.$

######  Remark 3.17.

The bound on the constant in this lemma is not optimal. Notice also that this
result entails $\rho_{\varepsilon}\to\infty$ as $\varepsilon\to 0$.

######  Lemma 3.18.

Keeping the same notation (cf. notation 3.13) as before,

$U^{\varepsilon}\xrightarrow[\varepsilon\to 0]{L^{r}}0\quad\text{and}\quad
U^{\varepsilon}\xrightarrow[\varepsilon\to 0]{\text{a.s.}}0\,.$

for every $r\geq 1$.

######  Lemma 3.19.

For every $k$, there exists a constant $D_{k}$ such that

$1-D_{k}e^{-\lambda}\mathbb{E}\\!\left[U^{\varepsilon}\right]^{k}\lesssim\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]\leq 1\quad\text{as }\varepsilon\to 0\,.$

Furthermore, for real $\gamma$ and $\sigma$,

$1-\mathbb{E}\\!\left[e^{-\gamma U^{\varepsilon}}\right]\leq\left\lvert
1-\mathbb{E}\\!\left[e^{-(\gamma+i\sigma)U^{\varepsilon}}\right]\right\rvert\,,$

In particular,

$\left\lvert\frac{\mathbb{E}\\!\left[e^{-(\gamma+i\sigma)U^{\varepsilon}}\right]}{1-\mathbb{E}\\!\left[e^{-(\gamma+i\sigma)U^{\varepsilon}}\right]}\right\rvert\leq\frac{\mathbb{E}\\!\left[e^{-\gamma
U^{\varepsilon}}\right]}{1-\mathbb{E}\\!\left[e^{-\gamma
U^{\varepsilon}}\right]}\,.$

######  Lemma 3.20.

For any $\delta>0$, weakly in $L^{2}([\delta,\infty[)$, for any $k\geq 0$

$\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}e^{\lambda
t}\lambda^{k}\;d\lambda\xrightarrow[T\to\infty]{\mathcal{D}}0$

at a rate $O(T^{-n})$ for any $n\in\mathbb{N}$. Furthermore, for any $k\geq 0$

$\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\frac{e^{\lambda
t}}{\lambda^{k}}\;d\lambda\xrightarrow[T\to\infty]{\mathcal{D}}\frac{t^{k-1}}{(k-1)!}$

weakly at a rate $O(T^{-(n+k-1)})$ and the convergence is strong as soon as
$k\geq 2$.

###### Proof of lemma 3.16.

It is sufficient to show it for $S^{\varepsilon}$ knowing that an analogous
treatment is possible for $T^{\varepsilon}$. The points in
$[0,S^{\varepsilon}]$ are characterized by the fact that

$X_{t}-\inf_{[0,t]}X<\varepsilon\,.$

In particular, the supremum over all $t$ ranging within $[0,S^{\varepsilon}]$
of this quantity is also less than $\varepsilon$.

$\mathbb{P}(S^{\varepsilon}>a)=\mathbb{P}\\!\left[\sup_{0\leq t\leq
a}\left(X_{t}-\inf_{[0,t]}X\right)<\varepsilon\right]\,.$

Consider now an interval $[0,k\mu]$, where $\mu>0$ and $k$ is an integer and
let us slice this interval into $k$ segments of length $\mu$. It is clear that
the following inequality holds

$\sup_{0\leq t\leq k\mu}\left(X_{t}-\inf_{[0,t]}X\right)\geq\sup_{1\leq j\leq
k}\left(\sup_{(j-1)\mu\leq t\leq
j\mu}\left(X_{t}-\inf_{[(j-1)\mu,t]}X\right)\right)\,,$

since over each smaller interval we can have only smaller spread than over the
entire interval. However, the right hand side is a supremum over i.i.d. random
variables, so that

$\displaystyle\mathbb{P}(S^{\varepsilon}>k\mu)$
$\displaystyle\leq\mathbb{P}\\!\left[\sup_{1\leq j\leq
k}\left(\sup_{(j-1)\mu\leq t\leq
j\mu}\left(X_{t}-\inf_{[(j-1)\mu,t]}X\right)\right)<\varepsilon\right]$
$\displaystyle=\mathbb{P}\\!\left[\sup_{0\leq
t\leq\mu}\left(X_{t}-\inf_{[0,t]}X\right)<\varepsilon\right]^{k}=\mathbb{P}(S^{\varepsilon}>\mu)^{k}\,.$

By the non-monotonicity of $X$, $\mathbb{P}(S^{\varepsilon}>\mu)<1$. In
particular, if we let $\mu=1$, and denote $c=\mathbb{P}(S^{\varepsilon}>1)<1$,
then:

$\lim_{k\to\infty}e^{\lambda
k}\,\mathbb{P}(S^{\varepsilon}>k)\leq\lim_{k\to\infty}e^{\lambda
k}\,\mathbb{P}(S^{\varepsilon}>1)^{k}=\lim_{k\to\infty}(e^{\lambda}c)^{k}=0$

as soon as $\lambda<\log(1/c)$. It follows that
$\mathbb{E}\\!\left[e^{-\lambda S^{\varepsilon}}\right]$ is well-defined for
$\lambda$ in some neighbourhood of zero and in particular all moments of
$S^{\varepsilon}$ are well-defined and finite. Finally, combining the above
remark with an application of Markov’s inequality we get

$\mathbb{P}(S^{\varepsilon}>2k\mathbb{E}\\!\left[S^{\varepsilon}\right])\leq
2^{-k}\,.$

Almost surely, $\frac{S^{\varepsilon}}{2\mathbb{E}[S^{\varepsilon}]}\leq G$,
where $G$ is a geometric random variable. The moments of $G$ can easily be
calculated, yielding the estimation in the lemma for the moments..

This shows that the radius of convergence of the Taylor series
$\mathbb{E}\\!\left[e^{-\lambda S^{\varepsilon}}\right]$ is bounded below by
$-\log(\mathbb{P}(S^{\varepsilon}>1))$, since Taylor series converge up to
their nearest singularity. If $X$ is $\alpha$-stable, we can use the relation
which tells us that, in distribution $U^{\varepsilon}=\varepsilon^{\alpha}U$,
so that by the ratio test,

$\limsup_{k\to\infty}\frac{\left\lvert\lambda\right\rvert}{k+1}\;\frac{\mathbb{E}\\!\left[(U^{\varepsilon})^{k+1}\right]}{\mathbb{E}\\!\left[(U^{\varepsilon})^{k}\right]}=\varepsilon^{\alpha}\left\lvert\lambda\right\rvert\limsup_{k\to\infty}\frac{1}{k+1}\;\frac{\mathbb{E}\\!\left[U^{k+1}\right]}{\mathbb{E}\\!\left[U^{k}\right]}\,.$

This limit is equal to $0\leq\rho^{-1}<\infty$, since
$\mathbb{E}\\!\left[e^{-\lambda U}\right]$ is analytic on a non-trivial disk
around $0$, by lemma 3.16 and usual properties of the Laplace transform.

$\limsup_{k\to\infty}\frac{\left\lvert\lambda\right\rvert}{k+1}\;\frac{\mathbb{E}\\!\left[(U^{\varepsilon})^{k+1}\right]}{\mathbb{E}\\!\left[(U^{\varepsilon})^{k}\right]}<1\,,$

whenever $\left\lvert\lambda\right\rvert<\rho\varepsilon^{-\alpha}$, which
shows the desired result.

∎

###### Proof of lemma 3.18.

The statement in $L^{r}$ follows from the following observation.

$0\leq\mathbb{E}\\!\left[(U^{\varepsilon})^{r}\right]\leq\sum_{k=1}^{\infty}k^{r}\mathbb{P}(U^{\varepsilon}\geq
k)\leq\sum_{k=1}^{\infty}k^{r}\mathbb{P}(U^{\varepsilon}\geq 1)^{k}$

by the arguments of lemma 3.16. This sum converges, since
$\mathbb{P}(U^{\varepsilon}\geq 1)<1$. As $\varepsilon\to 0$,
$\mathbb{P}(U^{\varepsilon}>1)\to 0$, since $X$ is almost surely nowhere
monotone, so that the entire sum tends to $0$. The almost sure statement
follows from the fact that $U^{\varepsilon}$ is monotone, since both
$T^{\varepsilon}$ and $S^{\varepsilon}$ are monotone functions of
$\varepsilon$. Since $L^{r}$ convergence implies almost sure convergence along
a subsequence $\varepsilon_{n}$, for
$\varepsilon_{n+1}<\varepsilon<\varepsilon_{n+1}$ by monotonicity of
$U^{\varepsilon}$

$U^{\varepsilon_{n+1}}<U^{\varepsilon}<U^{\varepsilon_{n}}\,,$

so the convergence is almost sure. ∎

###### Proof of lemma 3.19.

The first inequality of the lemma relies on the fact that

$\displaystyle\mathbb{E}\\!\left[e^{-\lambda T^{\varepsilon}}\right]$
$\displaystyle\leq\sum_{k\geq 0}e^{-\lambda
k}\mathbb{P}(T^{\varepsilon}>k)\leq\sum_{k\geq
0}\left[e^{-\lambda}\mathbb{P}(T^{\varepsilon}>1)\right]^{k}$
$\displaystyle=\frac{1}{1-e^{-\lambda}\mathbb{P}(T^{\varepsilon}>1)}\sim
1-e^{-\lambda}\mathbb{P}(T^{\varepsilon}>1)\quad\text{as }\varepsilon\to 0\,,$

since $\mathbb{P}(T^{\varepsilon}>1)\xrightarrow{\varepsilon\to 0}0$. Notice
an analogous inequality holds for $S^{\varepsilon}$. By Markov’s inequality,
we know that

$\mathbb{P}(T^{\varepsilon}>1)\leq\mathbb{E}\\!\left[(T^{\varepsilon})^{k}\right]\leq
C_{k}\mathbb{E}\\!\left[T^{\varepsilon}\right]^{k}$

from which the first inequality follows by lemma 3.18. The second and third
inequalities follow from noticing that for any $x$ and $y$

$\left\lvert\left\lvert x\right\rvert-\left\lvert
y\right\rvert\right\rvert\leq\left\lvert x-y\right\rvert$

and applying Jensen’s inequality. ∎

###### Proof of lemma 3.20.

Consider a test function $\varphi\in C_{c}^{\infty}([\delta,\infty[)$, then
integrating by parts

$\displaystyle\frac{1}{2\pi i}\int_{\mathbb{R}}\left[\int_{\gamma-
iT}^{\gamma+iT}e^{\lambda t}\lambda^{k}\;d\lambda\right]\varphi(t) dt$
$\displaystyle=\frac{(-1)^{k}}{2\pi
i}\int_{\mathbb{R}}dt\;\varphi^{(k)}(t)\int_{\gamma-iT}^{\gamma+iT}e^{\lambda
t}\;d\lambda$ $\displaystyle=(-1)^{k}\int_{\mathbb{R}}\frac{e^{\gamma
t}\varphi^{(k)}(t)}{\pi t}\sin(Tt)\;dt$ (3.5)

By performing the change of variables $y=Tt$, we see that the integral is
weakly approaching $0$, as $\varphi$ is not supported at $0$. Additionally,
away from $0$, the function

$\frac{e^{\gamma t}\varphi^{(k)}(t)}{\pi t}$

is a compactly supported $C^{\infty}$-function, integrating by parts $n$
subsequent times equation 3.5 yields bounds of this integral by
$C_{\varphi}T^{-n}$, where $C_{\varphi}$ is a constant which depends on the
test function and its support.

Let us now show that

$\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\frac{e^{\lambda
t}}{\lambda^{k}}\;d\lambda\xrightarrow[T\to\infty]{\mathcal{D}}\frac{t^{k-1}}{(k-1)!}\,.$

Once again integrating by parts,

$\displaystyle\int_{\mathbb{R}}dt\;\varphi(t)\left[\frac{1}{2\pi
i}\int_{\gamma-iT}^{\gamma+iT}\frac{e^{\lambda
t}}{\lambda^{k}}\;d\lambda-\frac{t^{k-1}}{(k-1)!}\right]$
$\displaystyle=(-1)^{n}\int_{\mathbb{R}}dt\;\varphi^{(n)}(t)\left[\frac{1}{2\pi
i}\int_{\gamma-iT}^{\gamma+iT}\frac{e^{\lambda
t}}{\lambda^{k+n}}\;d\lambda-\frac{t^{n+k-1}}{(n+k-1)!}\right]\,.$

Applying the residue theorem to evaluate the complex integral we get, for
$T>\gamma$,

$\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\frac{e^{\lambda
t}}{\lambda^{k+n}}\;d\lambda=\frac{t^{n+k-1}}{(n+k-1)!}+\frac{1}{2\pi
i}\int_{C_{T}}\frac{e^{\lambda t}}{\lambda^{k+n}}\;d\lambda\;,$

where $C_{T}$ is the circle of center $\lambda=\gamma$ and radius $T$. By the
estimation lemma, the contribution of this integral is bounded by $e^{\gamma
t}T^{-(n+k-1)}$ . It follows that

$\left\lvert\int_{\mathbb{R}}dt\;\varphi(t)\left[\frac{1}{2\pi i}\int_{\gamma-
iT}^{\gamma+iT}\frac{e^{\lambda
t}}{\lambda^{k}}\;d\lambda-\frac{t^{k-1}}{(k-1)!}\right]\right\rvert\leq
T^{-(n+k-1)}\left\lVert e^{\gamma t}\varphi^{(n)}(t)\right\rVert_{L^{1}}\,,$

thereby giving the speed of convergence desired. ∎

###### Proof of theorem 3.15.

Throughout this proof, we shall denote

$F(\lambda,\varepsilon):=\frac{1}{\lambda}\,\frac{\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]}{1-\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]}\,.$

The assumption of non-monotonicity of the Lévy process ensures that, almost
surely, $S^{\varepsilon}$ and $T^{\varepsilon}$ both tend to $0$ as
$\varepsilon\to 0$. Consider now the times $T_{i}^{\varepsilon}$ and
$S_{i}^{\varepsilon}$ given in definition 3.36. Since $X$ is Lévy,
$T_{i+1}^{\varepsilon}-S_{i}^{\varepsilon}$ and
$S_{i}^{\varepsilon}-T_{i}^{\varepsilon}$ are independent from one another,
and are both equal in distribution to $T^{\varepsilon}$ and $S^{\varepsilon}$
respectively.

By lemma 3.16, $S^{\varepsilon}$ and $T^{\varepsilon}$ admit finite moments
for all $k$ and the function $\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]$ is well defined, so, by equation 3.3

$\displaystyle{\mathcal{L}}(\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right])(\lambda)$
$\displaystyle={\mathcal{L}}(\mathbb{P}(R_{t}\geq\varepsilon))(\lambda)+\frac{1}{\lambda}\sum_{k\geq
1}\mathbb{E}\\!\left[e^{-\lambda U^{\varepsilon}}\right]^{k}$
$\displaystyle={\mathcal{L}}(\mathbb{P}(R_{t}\geq\varepsilon))(\lambda)+\frac{1}{\lambda}\frac{1}{\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]^{-1}-1}$ (3.6)

If we denote $\rho_{\varepsilon}$ the radius of convergence of the Taylor
series at zero associated to $\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]$, for
$\left\lvert\lambda\right\rvert<\rho_{\varepsilon}$,

$\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}\mathbb{E}\\!\left[(U^{\varepsilon})^{k}\right]}{k!}\,.$

This radius of convergence $\rho_{\varepsilon}$ can be bounded below with the
results of lemma 3.16 by

$-\log(\mathbb{P}(S^{\varepsilon}>1)\vee\mathbb{P}(T^{\varepsilon}>1))<\rho_{\varepsilon}\,,$

which entails that $\rho_{\varepsilon}\to\infty$ as $\varepsilon\to 0$. We
deduce from this series the Laurent series associated to
$\lambda^{-1}(\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]^{-1}-1)^{-1}$, namely

$\displaystyle
F(\lambda,\varepsilon)=\frac{1}{\lambda^{2}\mathbb{E}\\!\left[U^{\varepsilon}\right]}$
$\displaystyle+\frac{1}{\lambda}\left[\frac{\mathbb{E}\\!\left[(U^{\varepsilon})^{2}\right]}{2\mathbb{E}\\!\left[U^{\varepsilon}\right]^{2}}-1\right]+\frac{3\mathbb{E}\\!\left[(U^{\varepsilon})^{2}\right]^{2}-2\mathbb{E}\\!\left[U^{\varepsilon}\right]\mathbb{E}\\!\left[(U^{\varepsilon})^{3}\right]}{12\mathbb{E}\\!\left[U^{\varepsilon}\right]^{3}}+O(\lambda)\,.$

where the remainder in $\lambda$ is an analytic function of $\lambda$ for
$\left\lvert\lambda\right\rvert<\rho_{\varepsilon}$. By the inequalities of
lemma 3.19, the function doesn’t admit any poles on the half plane
$\operatorname{Re}(\lambda)>0$, so that the Taylor series above converges over
the same disk as that of $\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]$.

Observe now that for some small $\gamma>0$, the inverse Laplace transform of
$F(\lambda,\varepsilon)$ can be written as

${\mathcal{L}}^{-1}[F](t,\varepsilon)=\frac{1}{2\pi
i}\left\\{\int_{\gamma-i\rho_{\varepsilon}}^{\gamma+i\rho_{\varepsilon}}+\int_{\gamma+i\rho_{\varepsilon}}^{\gamma+i\infty}+\int_{\gamma-i\infty}^{\gamma-i\rho_{\varepsilon}}\right\\}e^{\lambda
t}F(\lambda,\varepsilon)\;d\lambda\,.$ (3.7)

Weakly, the integrals going off to infinity are of order
$o(\rho_{\varepsilon}^{-n})$ for any $n\in\mathbb{N}$, since for any test
function $\varphi\in C_{c}^{\infty}([0,\infty[)$, integrating by parts

$\int_{\mathbb{R}}dt\;\varphi(t)\left[\frac{1}{2\pi
i}\int_{\gamma+i\rho_{\varepsilon}}^{\gamma+i\infty}e^{\lambda
t}F(\lambda,\varepsilon)\;d\lambda\right]=\frac{(-1)^{n}}{2\pi
i}\int_{\mathbb{R}}dt\;\varphi^{(n)}(t)\int_{\gamma+i\rho_{\varepsilon}}^{\gamma+i\infty}d\lambda\;\frac{e^{\lambda
t}}{\lambda^{n}}F(\lambda,\varepsilon)\;d\lambda$

But using lemma 3.19,

$\left\lvert\int_{\gamma+i\rho_{\varepsilon}}^{\gamma+i\infty}d\lambda\;\frac{e^{\lambda
t}}{\lambda^{n}}F(\lambda,\varepsilon)\;d\lambda\right\rvert\leq e^{\gamma
t}\int_{\gamma+i\rho_{\varepsilon}}^{\gamma+i\infty}\left\lvert\frac{F(\lambda,\varepsilon)}{\lambda^{n}}\right\rvert\;d\lambda=O(\rho_{\varepsilon}^{-n-2})\,,$

which entails that the integrals going to infinity in equation 3.7 converge
weakly to $0$ at a rate $o(\rho_{\varepsilon}^{-n})$ for any $n\in\mathbb{N}$.
Thus, asymptotically as $\varepsilon\to 0$, for $t>0$,

$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]=\frac{t}{\mathbb{E}\\!\left[U^{\varepsilon}\right]}+\left(\frac{\mathbb{E}\\!\left[(U^{\varepsilon})^{2}\right]}{2\mathbb{E}\\!\left[U^{\varepsilon}\right]^{2}}-1\right)+\mathbb{P}(R_{t}\geq\varepsilon)+o(\rho_{\varepsilon}^{-n})\,,$

for any $n\in\mathbb{N}$. If the process is $\alpha$-stable, then
$(X_{c^{\alpha}t})_{t\geq 0}=(cX_{t})_{t\geq 0}$ in distribution for all $c$,
so that $U^{\varepsilon}=\varepsilon^{\alpha}U$ in distribution and

$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]=\frac{t}{\mathbb{E}\\!\left[U\right]\varepsilon^{\alpha}}+\frac{\mathbb{E}\\!\left[U^{2}\right]}{2\mathbb{E}\\!\left[U\right]^{2}}+o(\varepsilon^{\alpha
n})\quad\text{as }\varepsilon\to 0\,,$

for all $n\in\mathbb{N}$. As $\varepsilon\to 0$,
$1-\mathbb{P}(R_{t}\geq\varepsilon)=o(\varepsilon^{n})$ for any $n$, since

$\mathbb{P}(R_{t}\leq\varepsilon)\leq\mathbb{P}(T^{\varepsilon}>t)\leq\frac{\varepsilon^{\alpha
k}\mathbb{E}\\!\left[(T^{1})^{k}\right]}{t^{k}}$

for any $k$ by Markov’s inequality. ∎

######  Remark 3.21.

A similar theorem can be proven in $L^{s}(\Omega)$ for $\alpha$-stable
processes. For instance, if $X$ is $\alpha$-stable and $s=2$ one obtains that
for every $n\in\mathbb{N}$,

$\displaystyle\text{Var}(N^{\varepsilon}_{t})$
$\displaystyle\sim\left[\frac{\text{Var}(U)-2\mathbb{E}\\!\left[U\right]^{2}}{\mathbb{E}\\!\left[U\right]^{3}}\right]\frac{t}{\varepsilon^{\alpha}}$
$\displaystyle+\frac{5\text{Var}(U)^{2}}{4\mathbb{E}\\!\left[U\right]^{4}}+\frac{\text{Var}(U)}{\mathbb{E}\\!\left[U\right]^{2}}-\frac{2\mathbb{E}\\!\left[U^{3}\right]}{3\mathbb{E}\\!\left[U\right]^{3}}+\frac{7}{4}+o(\varepsilon^{\alpha
n})\quad\text{as }\varepsilon\to 0\,.$

Interestingly, there is a constant term appearing in this expansion, which can
be understood as induced by the boundary. This interpretation comes from
Picard’s analysis of the problem [37], in which the first term of this
asymptotic series was also derived (cf. proposition 3.11).

If $X$ has almost surely discontinuous paths, $X_{t}$ exhibits macroscopic
jumps. These will turn out to bring significative contributions, so much so
that

######  Corollary 3.22.

If $\alpha\neq 2$, the $\zeta$-function of any $\alpha$-stable Lévy process is
ill-defined for any $p\in\mathbb{C}$.

###### Proof.

The $\zeta$-function of a stochastic process $X$ can be written as

$\zeta_{X}(p)=\mathbb{E}\\!\left[R^{p}_{t}\right]+\mathbb{E}\\!\left[\sum_{k\geq
2}\ell_{k}^{p}(X_{t})\right]\,,$ (3.8)

if $X$ is $\alpha$-stable, the first term can be written as

$\mathbb{E}\\!\left[R_{t}^{p}\right]=t^{\frac{p}{\alpha}}\mathbb{E}\\!\left[R_{1}^{p}\right]\geq
t^{\frac{p}{\alpha}}\mathbb{E}\\!\left[\left\lvert
X_{1}\right\rvert^{p}\right]\,,$

where we have momentarily taken $p\in\mathbb{R}$. Since $X_{1}$ has a Lévy
$\alpha$-stable distribution, taking $p$ now complex,
$\mathbb{E}\\!\left[R_{t}^{p}\right]$ is infinite as soon as
$\operatorname{Re}(p)\geq\alpha$, since $X_{1}$ does not admit any moments of
order (of real part) larger than $\alpha$. Applying theorem 3.15, we know that
the second term in the above decomposition of $\zeta_{X}$ is only defined for
$\operatorname{Re}(p)>\alpha$, so the fundamental strip of
${\mathcal{M}}\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]$ is empty. ∎

In fact, it is possible to show that
$\mathbb{P}(X_{1}>\varepsilon)\sim\mathbb{P}(R_{1}>\varepsilon)$ as
$\varepsilon\to\infty$. It turns out that the distribution of $R_{1}$ is
dominated by the probability of having one large jump, which confirms our
previous statement on the effect of the discontinuity of Lévy processes on the
distribution of $R$. This is the so-called single big jump principle.

######  Proposition 3.23 (Single big jump principle, Bertoin, [5]).

If $X$ is an $\alpha$-stable process ($\alpha<2$), there exists a constant $k$
such that

$\mathbb{P}(R_{1}\geq\varepsilon)\sim\frac{k}{\varepsilon^{\alpha}}\quad\text{as
}\varepsilon\to\infty\,.$

Loosely speaking, it is intuitive to think that a corrective asymptotic power
series for $\mathbb{P}(R_{t}\geq\varepsilon)$ of the form

$\mathbb{P}(R_{1}\geq\varepsilon)\sim\sum_{k\geq
1}a_{k}\varepsilon^{-k\alpha}\quad\text{ as }\varepsilon\to\infty$

should exist for the following reason. By the single big jump principle, the
probability that the range exceeds $\varepsilon$ for large $\varepsilon$ is
dominated by the probability of a single big jump. However, it is also
possible to have $n$ large jumps of size $J_{k}\varepsilon$ where
$\sum_{k}^{n}J_{k}\geq 1$. The probability of each of these jumps happening is
of order $O(\varepsilon^{-\alpha})$ and by independence, the probability that
$k$ jumps of size $O(\varepsilon)$ happen is $O(\varepsilon^{-\alpha k})$. In
general, we cannot expect these events to be disjoint from one another, so the
coefficients $a_{k}$ of this sum may be negative. Finally, by the scaling
invariance it is sufficient to show that this is so for $R_{1}$. Corrective
terms to the above asymptotic relation should thus in principle exist, but the
explicit calculation of these terms is out of the scope of this paper.

By contrast, we will now show that
$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]-\mathbb{P}(R_{t}\geq\varepsilon)$
is well-behaved. This motivates the following definition

######  Definition 3.24.

The tail $\zeta$-function of the stochastic process $X$ on $[0,t]$ is defined
as

$\hat{\zeta}_{X}(p):=\mathbb{E}\\!\left[\operatorname{Pers}_{p}^{p}(X)-R^{p}_{t}\right]\,.$

######  Theorem 3.25.

The tail $\zeta$-function associated to an $\alpha$-stable Lévy process is
given by

$\hat{\zeta}_{X}(p)=\frac{t^{\frac{p}{\alpha}}}{\Gamma(\frac{p}{\alpha})}B^{*}\\!\left(\frac{p}{\alpha}\right)\,,$
(3.9)

which extends to a meromorphic function of $p$ to the entire complex plane
(since $B^{*}$ is itself meromorphic), with a unique simple pole at $p=\alpha$
of residue $\mathbb{E}\\!\left[U\right]^{-1}\alpha t$.

###### Proof of theorem 3.25.

To show that this quantity is well-defined, let us start by noticing that

${\mathcal{L}}(\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]-\mathbb{P}(R_{t}\geq\varepsilon))(\lambda)=\frac{\mathbb{E}\\!\left[e^{-\lambda\varepsilon^{\alpha}U}\right]}{\lambda(1-\mathbb{E}\\!\left[e^{-\lambda\varepsilon^{\alpha}U}\right])}$

which for $\operatorname{Re}(\lambda)>0$ goes to zero (uniformly in $\lambda$)
exponentially fast as $\varepsilon\to\infty$, showing that
$\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]-\mathbb{P}(R_{t}\geq\varepsilon)$
does as well for $\varepsilon\to\infty$ by an application of Markov’s
inequality. We can also compute the contribution of the second term of
equation 3.8. First, notice that

${\mathcal{L}}\\!\left(\mathbb{E}\\!\left[\sum_{k\geq
2}\ell_{k}^{p}(X_{t})\right]\right)\\!(\lambda)=\frac{p}{\lambda}{\mathcal{M}}\\!\left[\frac{\mathbb{E}\\!\left[e^{-\lambda\varepsilon^{\alpha}U}\right]}{1-\mathbb{E}\\!\left[e^{-\lambda\varepsilon^{\alpha}U}\right]}\right](p)$

Using the scaling property of the Mellin transform
${\mathcal{M}}_{z}[f(\lambda z)](p)=\lambda^{-p}f^{*}(p)$ and inverting the
Laplace transform

$\mathbb{E}\\!\left[\operatorname{Pers}_{p}^{p}(X)-R^{p}_{t}\right]=\frac{pt^{\frac{p}{\alpha}}}{\Gamma(1+\frac{p}{\alpha})}{\mathcal{M}}\\!\left[\frac{\mathbb{E}\\!\left[e^{-\varepsilon^{\alpha}U}\right]}{1-\mathbb{E}\\!\left[e^{-\varepsilon^{\alpha}U}\right]}\right](p)\,.$
(3.10)

Finally, setting

$B(z):=\frac{\mathbb{E}\\!\left[e^{-zU}\right]}{1-\mathbb{E}\\!\left[e^{-zU}\right]}\quad\text{and}\quad
B^{*}(p):={\mathcal{M}}_{z}[B(z)](p)\,,$

the polynomial scaling property of the Mellin transform,
${\mathcal{M}}_{z}[f(z^{\alpha})](p)=\frac{1}{\alpha}f^{*}(\frac{p}{\alpha})$
yields the final result. ∎

######  Remark 3.26.

Theorem 3.25 can be used to give an alternative proof for the series expansion
of theorem 3.15.

###### Alternate proof of theorem 3.15.

By lemma 3.16 and the analyticity of the expresion of $B$ with respect to
$\mathbb{E}\\!\left[e^{-zU}\right]$, $B$ admits a Laurent series on some non-
trivial annulus around zero with a single simple pole at $z=0$. By the
fundamental correspondence (theorem 2.5), the existence of this Laurent
expansion guarantees that $B^{*}(\frac{p}{\alpha})$ admits a meromorphic
continuation to the whole complex plane with only simple poles at every
$p=-n\alpha$ for every $n\in\mathbb{N}$ and at $p=\alpha$. The poles at the
negative integer multiples of $\alpha$ are compensated exactly by those of the
$\Gamma$-function in the denominator of the expression of $\hat{\zeta}$,
leaving only a pole at $\alpha$. Now, recalling that

$\hat{\zeta}(p)=p{\mathcal{M}}[\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]-\mathbb{P}(R_{t}\geq\varepsilon)](p)\,,$

${\mathcal{M}}[\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]-\mathbb{P}(R_{t}\geq\varepsilon)]$
has a supplementary pole at $p=0$. Admitting that $\hat{\zeta}(p)/p$ has the
decay condition to apply the fundamental correspondence by inverting the
Mellin transform we get the asymptotic relation desired. ∎

####  3.3.1 Exponential corrections

The fundamental correspondence is limited in that it allows us only to
describe $\mathbb{E}\\!\left[N^{\varepsilon}\right]$ asymptotically up to
terms smaller than any polynomial. However, in accordance to the discussion of
section 2.1.1, a finer study of the analytic properties of $\hat{\zeta}$ can
yield the superpolynomial corrections to our estimate, assuming that $B(z)$
admits a meromorphic extension to the whole complex plane. Using lemmata 2.6
and 2.8,

$\displaystyle\hat{\zeta}_{X}(p)=t^{\frac{p}{\alpha}}\Gamma\\!\left(1-\frac{p}{\alpha}\right)\sum_{z_{0}\in\mathcal{P}}\operatorname{Res}((-z)^{\frac{p}{\alpha}-1}B(z);z_{0})$
(3.11)
$\displaystyle{\mathcal{M}}(\mathbb{E}\\!\left[N^{\varepsilon}_{t}\right]-\mathbb{P}(R_{t}\geq\varepsilon))(p)=-\frac{t^{\frac{p}{\alpha}}\Gamma\\!\left(-\frac{p}{\alpha}\right)}{\alpha}\sum_{z_{0}\in\mathcal{P}}\operatorname{Res}((-z)^{\frac{p}{\alpha}-1}B(z);z_{0})\,.$
(3.12)

Recognizing that

${\mathcal{M}}_{z}\left[\frac{e^{z_{0}/z}}{z_{0}}\right](p)=-\Gamma(-p)\left(\frac{-1}{z_{0}}\right)^{1-p}\,,$

we may formally invert the Mellin transform if all the $z_{0}$’s are simple
poles to obtain the exponentially small corrections

$\mathbb{E}\\!\left[N_{t}^{\varepsilon}\right]-\mathbb{P}(R_{t}\geq\varepsilon)-\frac{t}{\mathbb{E}\\!\left[U\right]\varepsilon^{\alpha}}-\left[\frac{\mathbb{E}\\!\left[U^{2}\right]}{2\mathbb{E}\\!\left[U\right]^{2}}-1\right]\sim\sum_{z_{0}\in\mathcal{P}}\frac{e^{tz_{0}/\varepsilon^{\alpha}}}{\alpha
z_{0}}\operatorname{Res}(B(z);z_{0})\quad\text{as }\varepsilon\to 0\,.$

Generally, the poles are not simple so the corrective terms to this series
stem from residues of higher order poles (the corrections remain nonetheless
superpolynomially small as $\varepsilon\to 0$).

####  3.3.2 Distribution of the length of branches

The distribution of the length of the $k$th branch (in the sense of figure 6)
can be calculated. Recall that

$\mathbb{E}\\!\left[\ell_{k}^{p}(X)\right]=p{\mathcal{M}}[\mathbb{P}(N_{t}^{\varepsilon}\geq
k)](p)\,.$

For $k\geq 2$,

${\mathcal{L}}[\mathbb{E}\\!\left[\ell_{k}^{p}(X)\right]](\lambda)=\frac{p}{\lambda}{\mathcal{M}}\left[\mathbb{E}\\!\left[e^{-\lambda
U^{\varepsilon}}\right]^{k-1}\right](p)\,.$

If we suppose once again that $X$ is a Lévy $\alpha$-stable process, this can
be simplified to yield

$\mathbb{E}\\!\left[\ell_{k}^{p}(X)\right]=\frac{t^{\frac{p}{\alpha}}}{\Gamma(\frac{p}{\alpha})}\,{\mathcal{M}}\\!\left[\mathbb{E}\\!\left[e^{-\varepsilon
U}\right]^{k-1}\right]\\!\left(\frac{p}{\alpha}\right)\,,$

Taking the Mellin transform of a power is in general difficult. Inversion is
also in general complicated due to the presence of the $\Gamma$-function in
the denominator of the above expression. To remediate the first problem, we
can form the generating function yielding the distribution for the $k$th bar,

$G_{\alpha}(z;p):=\sum_{k\geq
2}\mathbb{E}\\!\left[\ell_{k}^{p}(X)\right]z^{k}=\frac{t^{\frac{p}{\alpha}}}{\Gamma(\frac{p}{\alpha})}\,{\mathcal{M}}\\!\left[\frac{z}{(z\mathbb{E}\\!\left[e^{-\varepsilon
U}\right])^{-1}-1}\right]\\!\left(\frac{p}{\alpha}\right)\,,$

which allows us to express

######  Proposition 3.27.

For $k\geq 2$, the distribution of the length of the $k$th longest branch is
characterized by its Mellin transform which is given by

$\mathbb{E}\\!\left[\ell_{k}^{p}(X)\right]=\frac{1}{k!}\left.\frac{\partial^{k}}{\partial
z^{k}}\right|_{z=0}G_{\alpha}(z;p)\,.$

Whenever convenient, the expression above can also be evaluated by considering
a circular contour of small enough radius $r$ around the origin $C_{r}$ and
evaluating

$\mathbb{E}\\!\left[\ell_{k}^{p}(X)\right]=\frac{1}{2\pi
i}\oint_{C_{r}}\frac{G_{\alpha}(z;p)}{z^{k+1}}\;dz\,.$

####  3.3.3 Statistical parameter testing for $\alpha$-stable processes and
perspectives

What we will aim to do in this section is to illustrate by example why
barcodes can be a robust statistical tools for parameter testing. Parameter
testing is a widely studied subject, notably for self-similar processes, where
the problem has been treated in dimension $1$ (a non-comprehensive list of
references is [16] and the references therein). A variety of different
methods, such as multi-scale wavelet analysis, have been used to produce these
results (although other methods such as the ones of [16] have also been used),
so our approach does not offer anything new in this respect. The interest of
our method lies in possible applications to higher dimensional random fields,
for which wavelet analysis is not an effective tool. A complete theoretical
framework for this would require the study of the trees of higher dimensional
random fields, which are out of the scope of this paper : instead, this
section acts as a proof of concept for the use of topological estimators and
their utility, by studying what happens in dimension 1.

In what follows, we will consider $X$ to be an $\alpha$-stable Lévy process,
of which we will aim to estimate the parameter $\alpha$. From proposition 3.11
we know that almost surely

$N^{\varepsilon}_{t}\sim Ct\varepsilon^{-\alpha}\quad\text{as }\varepsilon\to
0\,.$

In particular, given some sample we may compute the sampled value of
$N^{\varepsilon}_{t}$, which we will denote $\hat{N}^{\varepsilon}_{t}$
explicitly. A close inspection of the behaviour of the sample mean
$\overline{N}^{\varepsilon}_{t}$ should thus yield an estimation for the
parameter $\alpha$ of the process $X$.

######  Remark 3.28.

In fact, the same reasoning allows us to estimate the Hurst parameter $H$ of a
fractional Brownian motion (fBM), which also exhibits self-similarity. In this
case, the analogue of the asymptotic result of proposition 3.11 is [37, §3]

$\text{a.s.}\quad N^{\varepsilon}\sim
Ct\varepsilon^{-\frac{1}{H}}\quad\text{as }\varepsilon\to 0\,.$

More precisely, given a sample, our test consists in performing the following
steps.

  1. 1.

Sample $M$ paths of the stochastic process $X$ (for example at regular
intervals of size $\frac{1}{N}$ for some $N$) ;

  2. 2.

Compute the barcode of the sampled paths. To do this, first construct a
filtered simplicial complex (which is in this case nothing other than a chain
with $\\!\sim\\!N$ links) by taking each point to be a vertex of a complex and
joining adjacent sampling points with an edge. The filtration on this complex
is the value of the process at the edge (for an edge connecting vertex $a$ to
vertex $b$, the value of the filtration is $X_{a}\wedge X_{b}$). Finally, the
persistent homology of this complex can be computed with the gudhi package
[1], which incidentally also offers a convenient implementation of filtered
simplicial complexes due to Boissonnat and Maria [7].

  3. 3.

For some range of small enough $\varepsilon$, and for some positive constant
$c>1$ compute the quantity

$\hat{\alpha}_{M}:=\log_{c}\left[\frac{\overline{N}^{\varepsilon/c}_{t}-\overline{N}^{2\varepsilon/c}_{t}}{\overline{N}^{\varepsilon}_{t}-\overline{N}^{2\varepsilon}_{t}}\right]\,.$

Here, the notion of some range of small enough $\varepsilon$ and the constant
$c$ both depend on $N$, with the limiting condition that as $N\to\infty$, the
lower bound on the range of valid $\varepsilon$ goes to zero.

Our claim is that the computed quantity $\hat{\alpha}$ is a valid estimation
of the parameter $\alpha$ (for fBM, the quantity obtained in this way is an
estimate of $\frac{1}{H}$).

######  Lemma 3.29 (Convergence of the sample means).

The quotient

$\frac{\overline{N}^{\varepsilon/c}_{t}-\overline{N}^{2\varepsilon/c}_{t}}{\overline{N}^{\varepsilon}_{t}-\overline{N}^{2\varepsilon}_{t}}\xrightarrow[M\to\infty]{\mathbb{P}}\frac{\mathbb{E}\\!\left[N^{\varepsilon/c}_{t}-N^{2\varepsilon/c}_{t}\right]}{\mathbb{E}\\!\left[N^{\varepsilon}_{t}-N^{2\varepsilon}_{t}\right]}$

at a rate $C_{s}M^{-s}$, for every $1\leq s\leq 2$ where $C_{s}$ is a constant
depending on $s$ and the $s$th moment of $N^{\varepsilon/c}$. In particular,

$\hat{\alpha}_{M}\xrightarrow[M\to\infty]{\mathbb{P}}\alpha+\xi(\varepsilon)$

at the same rate, where $\xi(\varepsilon)$ is a superpolynomially small
function of $\varepsilon$.

######  Remark 3.30.

The at first seemingly convoluted expression for the estimator
$\hat{\alpha}_{M}$ can be explained due to the results of theorem 3.15. The
substraction present in the numerator and denominator is performed so that the
constant terms of equation 3.4 vanish. Ignoring the superpolynomial
contributions to this expression which remain small, we then have that the
argument inside the $\log$ of the estimator is roughly

$c^{\hat{\alpha}_{M}}\approx\frac{\frac{t}{\mathbb{E}\\!\left[U\right](\varepsilon/c)^{\alpha}}-\frac{t}{\mathbb{E}\\!\left[U\right](2\varepsilon/c)^{\alpha}}}{\frac{t}{\mathbb{E}\\!\left[U\right]\varepsilon^{\alpha}}-\frac{t}{\mathbb{E}\\!\left[U\right](2\varepsilon)^{\alpha}}}\approx
c^{\alpha}\,.$

With this in mind, let us now formally prove the statement of lemma 3.29.

###### Proof.

That the numerator and the denominator tend to the respective expected values
holds by a simple application of the weak law of large numbers, since
$N^{\varepsilon}_{t}$ is a random variable in $L^{s}$ for $s\geq 1$. The rate
of convergence of this limit can be obtained via a simple application of
Markov’s inequality, by noting first that the summands in the denominator tend
to their limits faster than those of the numerator, as the latter’s $s$th
moments are always larger than the former’s. From theorem 3.15, we see that
the limit can be expressed as

$\frac{\mathbb{E}\\!\left[N^{\varepsilon/c}_{t}-N^{2\varepsilon/c}_{t}\right]}{\mathbb{E}\\!\left[N^{\varepsilon}_{t}-N^{2\varepsilon}_{t}\right]}=c^{\alpha}\;\frac{1+g(c\varepsilon)}{1+g(\varepsilon)}\,,$

where $g$ is a function tending to $0$ superpolynomially fast as
$\varepsilon\to 0$, determined by the superpolynomial corrections to the
results of theorem 3.15. The statement of the lemma ensues. ∎

######  Lemma 3.31 (Probable $L^{\infty}$-distance of sampling).

Denote $\hat{X}$ the trajectory samples of the $\alpha$-stable process $X$ at
every interval of length $\frac{1}{N}$. More precisely, somewhat abusing the
notation we can write,

$\hat{X}_{t}=\sum_{n=0}^{N-1}1_{[\frac{n}{N},\frac{n+1}{N}[}(t)\;X_{\frac{n}{N}}\,.$

There exists a constant $k$ such that

$\mathbb{P}\left(\sup_{t\in\mathbb{R}}\left\lvert
X_{t}-\hat{X}_{t}\right\rvert\leq\varepsilon\right)\gtrsim
1-\left(\frac{k}{N\varepsilon^{\alpha}}\right)^{N}\quad\text{ as }\varepsilon
N^{1/\alpha}\to\infty\,.$

######  Remark 3.32.

The asymptotic dependence above fixes admissible values of $\varepsilon$ as a
function of $N$ as holding whenever the asymptotic dependence above is valid
(it must be valid between $\varepsilon/c$ and $2\varepsilon$). Furthermore,
the parameter $c$ we chose above is also further constrained by the
requirement that the asymptotic relation of theorem 3.15 holds between
$\varepsilon/c$ and $2\varepsilon$. More precisely, we fix $c$ and
$\varepsilon$ such that the superpolynomial contributions in the expansion of
theorem 3.15 are negligible with respect to the term in
$\varepsilon^{-\alpha}$ and by imposing that $\varepsilon N^{1/\alpha}$ is
large enough so that the asymptotic relation of lemma 3.31 simultaneously
holds within the range $[\frac{\varepsilon}{c},2\varepsilon]$. In practice, we
may fix $c$ and $\varepsilon$ by looking at a log-log chart of
$N^{\varepsilon}$, the regime of validity of $\varepsilon$ and the value of
$c$ become clear, as shown in figure 9.

######  Remark 3.33.

With respect to the barcode, linear interpolation between values of $X$ or the
consideration of the process $\hat{X}$ is equivalent.

######  Remark 3.34.

It is not a priori obvious that the event above is measurable. However,
continuity in probability and the a.s. existence of a càdlàg modification of
the process allows us to interpret this event to be a supremum over every
$t\in\mathbb{Q}$, rendering the event measurable.

###### Proof.

It suffices to show the result over the interval $[0,1]$. Noticing that the
sampling coincides with the value of the path at every $t=\frac{1}{N}$, it
suffices to evaluate the probability that over $N$ intervals of length
$\frac{1}{N}$ the real sampled path $X_{t}(\omega)$ (notice the absence of a
hat) strays away from the sampled path $\hat{X}_{t}(\omega)$. Focusing on a
single interval $[0,\frac{1}{N}]$, the single big jump principle 3.23 states
that there exists a constant $k$ such that this probability of straying away
in this interval is

$\mathbb{P}(R_{\frac{1}{N}}\geq\varepsilon)\sim\frac{k\varepsilon^{-\alpha}}{N}\quad\text{
as }\varepsilon N^{1/\alpha}\to\infty\,.$

By independence, over $N$ such intervals

$\mathbb{P}\left(\sup_{t\in\mathbb{R}}\left\lvert
X_{t}-\hat{X}_{t}\right\rvert\leq\varepsilon\right)\gtrsim
1-\left(\frac{k}{N\varepsilon^{\alpha}}\right)^{N}\sim 1\quad\text{ as
}\varepsilon N^{1/\alpha}\to\infty\,,$

as desired. ∎

Figure 9: In orange, a histogram of the number of bars of length
$\geq\varepsilon$, $N^{\varepsilon}$ found as a function of $\log\varepsilon$
from a simulation of a Lévy $1.2$-stable process as a random walk. In blue,
the function $C_{1.2}\varepsilon^{-1.2}$.

Now, let us recall the following theorem.

######  Theorem 3.35 (P, Theorem 3.1 [36]).

Let $\delta_{N}:=\left\lVert X-\hat{X}\right\rVert_{L^{\infty}}$, then there
exists a $\delta_{N}$-matching between the barcodes of $\hat{X}$ and $X$. In
particular, for any $\varepsilon\geq 2\delta_{N}$

$N_{X}^{\varepsilon+\delta_{N}}\leq N_{\hat{X}}^{\varepsilon}\leq
N_{X}^{\varepsilon-\delta_{N}}$

Moreover, if $\mathbb{E}\\!\left[N_{X}^{\varepsilon}\right]$ is continuous
with respect to $\varepsilon$, then

$N_{\hat{X}}^{\varepsilon}\xrightarrow[N\to\infty]{L^{1}}N_{X}^{\varepsilon}\quad\text{and}\quad
N_{\hat{X}}^{\varepsilon}\xrightarrow[n\to\infty]{\mathbb{P}}N_{X}^{\varepsilon}\,,$

which at fixed $N$ quantitatively translates to

$\mathbb{E}\\!\left[\left\lvert
N_{X}^{\varepsilon}-N_{\hat{X}}^{\varepsilon}\right\rvert\right]\leq
2\omega_{\varepsilon}(\delta_{N})\quad\text{and}\quad\mathbb{P}(\left\lvert
N_{X}^{\varepsilon}-N_{\hat{X}}^{\varepsilon}\right\rvert\geq
k)\leq\frac{2\omega_{\varepsilon}(\delta_{N})}{k}$

where $\omega_{\varepsilon}$ is the modulus of continuity of
$\mathbb{E}\\!\left[N_{X}^{\varepsilon}\right]$ on the interval
$[\varepsilon-\delta_{N},\varepsilon+\delta_{N}]$. Finally, the following
inequalities also hold

$\displaystyle N^{\delta_{n}}_{\hat{X}}\geq N^{2\delta_{N}}_{X}\quad\text{ and
}\quad N^{\delta_{N}}_{X}\geq N_{\hat{X}}^{2\delta_{N}}\;.$

This statement can be specialized given our two lemmas above. The theorem
provides bounds on $N^{\varepsilon}_{\hat{X}}$ provided that we know the value
of $\delta_{N}$, since if $\varepsilon$ is small enough, $N^{\varepsilon}_{X}$
has some almost sure asymptotic behaviour. On the other hand, by virtue of
lemma 3.31 we have a probable estimate of $\delta_{N}$, i.e. with probability
$q$, we may give a bound of $\delta_{N}$, rendering the statement
quantitative. The second part of the statement of theorem 3.35 provides bounds
on the $L^{1}$ distance between $N^{\varepsilon}_{\hat{X}}$ and
$N^{\varepsilon}_{X}$, provided that we know that
$\mathbb{E}\\!\left[N^{\varepsilon}_{X}\right]$ is continuous. This happens to
be the case for Brownian motion, as shown in [36]. Showing it in full
generality for Lévy processes requires a closer study of the range of Lévy
processes and the continuity of the inverse Mellin transform of
$\hat{\zeta}_{X}(p)/p$. However, for the purposes of the construction of our
statistical test, lemma 3.29 suffices, as it provides us with a quantitative
guarantee that the parameter $\alpha$ is well estimated by our estimator
$\hat{\alpha}_{M}$.

###  3.4 Propagators and local $\zeta$-functions

In dimension one, it is possible to use the total order of $\mathbb{R}$ and
count $N^{x,x+\varepsilon}$ by counting the number of times we go up from $x$
to $x+\varepsilon$. This idea can be formalized by the following sequence of
stopping times already introduced in the literature of classical probability
theory (cf. for instance [41]).

######  Definition 3.36.

Setting $S_{0}^{x,\varepsilon}=T_{0}^{x,\varepsilon}=0$, we define a sequence
of times recursively

$\displaystyle T_{i+1}^{x,\varepsilon}$ $\displaystyle:=\inf\left\\{\,t\geq
S_{i}^{x,\varepsilon}\;\Bigg{|}\;f(t)\leq x\wedge(x+\varepsilon)\right\\}$
$\displaystyle S_{i+1}^{x,\varepsilon}$ $\displaystyle:=\inf\left\\{\,t\geq
T_{i+1}^{x,\varepsilon}\;\Bigg{|}\;f(t)\geq x\vee(x+\varepsilon)\right\\}\,.$
(3.13)

Counting the number of bars of length $\varepsilon$ is thus exactly to count
the number of up and downs we make. More precisely,

$N^{x,x+\varepsilon}=\inf\\{i\,|\,T_{i}^{x,\varepsilon}\text{ or
}S_{i}^{x,\varepsilon}=\inf\emptyset\\}\,,$ (3.14)

by which we mean that it is the smallest $i$ such that the set over which
$T_{i}^{\varepsilon}$ or $S_{i}^{\varepsilon}$ are defined as infima is empty.

With the aforementioned, it is possible to define a Feynman-like formalism to
perform the computation of $\mathbb{E}\\!\left[N^{x,x+\varepsilon}\right]$.

######  Definition 3.37.

Let $X$ be a (strong Markov) stochastic process and let

$T^{a}:=\inf\\{t\geq 0\,\mid\,X_{t}>a\\}\,$

be the hitting time of $a$ by $X$. We define the propagator from $x$ to $y$ by

$\langle x|y\rangle:=\mathbb{E}_{x}\\!\left[e^{-\lambda T^{y}}\right]\,$

whenever this exists.

######  Remark 3.38.

Whenever convenient, we may take $\lambda=i\omega$, and modify the subsequent
expressions appropriately.

If the process $X$ has the strong Markov property and the increments between
the stopping times $T_{i}^{x,\varepsilon}$ and $S_{i}^{x,\varepsilon}$ are
identically distributed, we can once again apply renewal theory. In
particular, the Laplace transform of the occupation numbers
${\mathcal{L}}N^{x,x+\varepsilon}_{t}$ can be understood in terms of the
propagators above. For $x>0$,

$\displaystyle{\mathcal{L}}(\mathbb{E}\\!\left[N^{x,x+\varepsilon}_{t}\right])(\lambda)$
$\displaystyle=\frac{\langle 0|x+\varepsilon\rangle}{\lambda}\sum_{k\geq
0}(\langle x+\varepsilon|x\rangle\langle x|x+\varepsilon\rangle)^{k}$
$\displaystyle=\frac{\langle 0|x+\varepsilon\rangle}{\lambda(1-\langle
x+\varepsilon|x\rangle\langle x|x+\varepsilon\rangle)}$

Similarly, all moments of the distribution can in principle be calculated

$\displaystyle{\mathcal{L}}(\mathbb{E}\\!\left[(N^{x,x+\varepsilon}_{t})^{s-1}\right])(\lambda)=\frac{\langle
0|x+\varepsilon\rangle(1-\langle x+\varepsilon|x\rangle\langle
x|x+\varepsilon\rangle)}{\lambda}\operatorname{Li}_{-s+1}(\langle
x+\varepsilon|x\rangle\langle x|x+\varepsilon\rangle)$

where $\operatorname{Li}$ denotes the polylogarithm.

######  Remark 3.39.

If $X_{t}-X_{s}=X_{t-s}$ in distribution, we can rewrite the above as,

${\mathcal{L}}(\mathbb{E}\\!\left[(N^{x,x+\varepsilon})^{s-1}\right])(\lambda)=\frac{\langle
0|x+\varepsilon\rangle(1-\langle\varepsilon|0\rangle\langle
0|\varepsilon\rangle)}{\lambda}\operatorname{Li}_{-s+1}(\langle\varepsilon|0\rangle\langle
0|\varepsilon\rangle)$

If $\langle x+\varepsilon|x\rangle\langle x|x+\varepsilon\rangle$ and $\langle
0|x+\varepsilon\rangle$ admit an asymptotic expansion for small $\varepsilon$