# Ladder costs for random walks
in Lévy random media

Alessandra Bianchi  Dipartimento di Matematica, Università di Padova, Via
Trieste 63, 35121 Padova, Italy<EMAIL_ADDRESS>,  Giampaolo
Cristadoro  Dipartimento di Matematica e Applicazioni, Università di Milano -
Bicocca, Via R. Cozzi 55 - 20125 Milano, Italy.
<EMAIL_ADDRESS>and  Gaia Pozzoli  Dipartimento di Scienza e
Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli
studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy, and Istituto
Nazionale di Fisica Nucleare - Sezione di Milano, Via Celoria 16, 20133
Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider a random walk $Y$ moving on a _Lévy random medium_ , namely a one-
dimensional renewal point process with inter-distances between points that are
in the domain of attraction of a stable law. The focus is on the
characterization of the law of the first-ladder height $Y_{\mathcal{T}}$ and
length $L_{\mathcal{T}}(Y)$, where $\mathcal{T}$ is the first-passage time of
$Y$ in ${\mathbb{R}}^{+}$. The study relies on the construction of a broader
class of processes, denoted _Random Walks in Random Scenery on Bonds_ (RWRSB)
that we briefly describe. The scenery is constructed by associating two random
variables with each bond of ${\mathbb{Z}}$, corresponding to the two possible
crossing directions of that bond. A random walk $S$ on ${\mathbb{Z}}$ with
i.i.d increments collects the scenery values of the bond it traverses: we
denote this composite process the RWRSB. Under suitable assumptions, we
characterize the tail distribution of the sum of scenery values collected up
to the first exit time $\mathcal{T}$. This setting will be applied to obtain
results for the laws of the first-ladder length and height of $Y$. The main
tools of investigation are a generalized Spitzer-Baxter identity, that we
derive along the proof, and a suitable representation of the RWRSB in terms of
local times of the random walk $S$. All these results are easily generalized
to the entire sequence of ladder variables.

MSC 2010: primary: 60G50, 60F05; secondary: 82C41, 60G55, 60F17.

Keywords: Spitzer identities, first-passage, random walk in random scenery,
stable distributions, Lévy-Lorentz gas

Acknowledgements: A. Bianchi is partially supported by the BIRD project
198239/19 _“Stochastic processes and applications to disordered systems”_
(UniPd). G. Cristadoro’s research is part of his activity within the Gruppo
Nazionale di Fisica Matematica(INdAM) and the UMI Group DinAmicI. G.
Cristadoro and G. Pozzoli are partially supported by the PRIN Grant 2017S35EHN
_“Regular and stochastic behavior in dynamical systems”_ (MIUR, Italy).

##  1\. Introduction

An essential component of fluctuation theory in discrete time is the study of
the first-ladder height and time of a one-dimensional (1D) random walk
$S=(S_{n})_{n\in{\mathbb{N}}_{0}}$, respectively given by the first maximal
value reached by $S$, and by the corresponding time. In this context, the
Wiener-Hopf techniques, introduced by Spitzer, Baxter and others, offer a main
tool of investigation, and allow for the derivation of several fundamental
identities that relate the distributions of these first-ladder random
variables to that of the underlying random walk (see [24, 25, 26], and [8, 10]
for reviews). These results, that have then been established in different
formulations by many authors, open the way to a refined understanding of the
first-ladder quantities of the ladder (ascending or descending) process, and
of conditioned random walks ([11, 12, 1, 15, 9]).

The aim of the present work is to generalize these kind of results to random
walks moving on a one-dimensional random medium having i.i.d. heavy-tailed
inter-distances. Specifically, the random medium that we consider is a renewal
point process $\omega=(\omega_{k})_{k\in{\mathbb{Z}}}$ with i.i.d. inter-
distances in the domain of attraction of a stable variable. The random walk on
the random medium $\omega$ is then defined as
$Y=(Y_{n})_{n\in{\mathbb{N}}_{0}}$ , where $Y_{n}:=\omega_{S_{n}}$ and
$S=(S_{n})_{n\in{\mathbb{N}}_{0}}$ is an underlying random walk on
${\mathbb{Z}}$, independent of $\omega$.

The process $Y$ can be seen as a generalization of the _Lévy flights_ , that
are random walks with i.i.d. heavy-tailed jumps, and as a discrete time
version of the _Lévy-Lorentz gas_ (see [5] and [6, 7] for some related
extensions). All these are processes that have been receiving a surge of
attention as they model phenomena of anomalous transport and anomalous
diffusion (see e.g. [16, 28, 3, 20, 29] for some general or recent
references).

From its definition, it turns out that the process $Y$ performs the same jumps
as $S$ but on the marked points of $\omega$ instead of ${\mathbb{Z}}$. Thus
the first-ladder times of $S$ and $Y$ correspond, and we set

$\mathcal{T}:=\min\\{n>0\,:\,S_{n}>0\\}\equiv\min\\{n>0\,:\,Y_{n}>0\\}\,.$

A complete characterization of the law of $\mathcal{T}$ can then be derived
from the classical Spitzer-Baxter identities, and specifically from the so
called Sparre-Andersen identity [22, 23] (see also [10] for a general
treatment, and [19] for the specific lattice case). In particular, if the
random walk $S$ has symmetric jumps as in our definition, then the law of
$\mathcal{T}$ is in the domain of attraction of a $1/2$-stable law.

The characterization of the distribution of $Y_{\mathcal{T}}$, the first-
ladder height of $Y$, is in general an open problem, as the double source of
randomness creates a non-trivial dependence between the increments of the
process and makes the analysis of the corresponding motion much harder than
the classical independent case, studied for example in [21, 12]. For the same
reason, the ascending ladder process
$(Y_{\mathcal{T}_{k}})_{k\in{\mathbb{N}}_{0}}$, where $\mathcal{T}_{k}$ is the
time corresponding to the $k$-th maximum value reached by $Y$ (so that
$\mathcal{T}_{0}=0$ and $\mathcal{T}_{1}\equiv\mathcal{T}$), is not in general
a renewal process.

We will approach the problem by considering a slightly generalized setting,
appearing in several applications, in which a cost process $C$ is associated
with a real (continuous or discrete) random walk $S$, that is assumed to be
the control process. As a simple but paradigmatic example, suppose that each
jump of the random walk takes a given and possibly random cost (e.g. time or
energy) to be performed. We could then be interested in the total cost
accumulated when the walk reaches its first maximum, that is the quantity
$C_{\mathcal{T}}$. As a first step of our analysis, we will derive a
generalized Spitzer-Baxter identity for $(\mathcal{T},C_{\mathcal{T}})$ under
the assumption that the process $C$ has i.i.d increments, eventually depending
on the control process (Theorem 2.1). When the cost process is chosen to be
exactly equal to $S$, we recover the classical Spitzer-Baxter identity. With
different choices of $C$, we are able to derive information on different kinds
of first-ladder random variables associated with the process, such as its
first-ladder length.

We then move to _Random Walks in Random Scenery on Bonds_. In this setting, as
already mentioned, at each step the walker collects the scenery values of the
bond it traverses. It is apparent that these quantities can be seen, in the
same spirit as the previous part, as the increments of a cost process $C$
associated with $S$. On the other hand such increments are now not i.i.d. and
thus Theorem 2.1 can not be directly applied. However, assuming that $S$ has
symmetric increments and thanks to a representation of the process in terms of
the local times of $S$, we will be able to express the generating function of
$C_{\mathcal{T}}$ in a simpler form, that allows for the implementation of the
generalized Spitzer-Baxter identities. The explicit results are derived from
Tauberian theorems under mild assumptions about the scenery process. Then we
will adapt the techniques used to analyze the first-ladder quantities to
obtain analogous results for the $k$-th ladder costs, $C_{\mathcal{T}_{k}}$.

Finally, these results are applied in order to derive the tail distributions
of the first-ladder length and height of the process $Y$. In turn, the latter
can be used to infer on the law of the first-passage time of a generalized
Lévy-Lorentz gas.

The paper is organized as follows. Section 2 is devoted to the rigorous
definition of cost and control processes, random walks in random scenery on
bonds and the related first-ladder quantities. At the same time, we provide
the statement of the associated main results. All the proofs of these theorems
are presented in Section 3, together with some explicit applications to random
walks in Lévy random media and Lévy-Lorentz gas.

##  2\. Setup and Main Results

We start by defining a convention that will be used throughout the paper. If
$X$ is a random variable in the domain of attraction of an $\alpha$-stable
law, with $\alpha\in(0,2]$, we define
$\hat{\alpha}=\textrm{min}\\{1,\alpha\\}$. If $X\equiv 0$, we set
$\hat{\alpha}=+\infty$.

Let us consider a process $S:=(S_{n})_{n\in{\mathbb{N}}_{0}}$ taking value on
${\mathbb{R}}$ and, for a fixed $\ell\in{\mathbb{N}}$, an $\ell$-dimensional
process $C:=(C_{n})_{n\in{\mathbb{N}}_{0}}$, that could depend non-trivially
on $S$. We denote by $(\xi_{k},\eta_{k})_{k\in\mathbbm{N}}$ the increments of
the joint process $(S,C)$. $C$ is referred to as the cost process while $S$ is
the control process111It is apparent that we can equivalently define an
$(\ell+1)$-dimensional process and choose an arbitrary coordinate to play the
role of the control process and the remaining $\ell$’s as the cost process. We
prefer to stick to a more explicit notation for the sake of clarity.. We avoid
to write explicitly the dependence on $S$ of the cost process, unless
explicitly needed.

Example 1. _As a simple but paradigmatic example to be used in next sections,
consider the one-dimensional cost process obtained by choosing
$\eta_{k}=|\xi_{k}|$, that is_

(2.1) $C_{n}(S)=\sum_{k=1}^{n}|S_{k}-S_{k-1}|\,=:L_{n}(S),\qquad\forall
n\in{\mathbb{N}}_{0}\,.$

_It is apparent that $L_{n}(S)$ measures the total length of the walker after
$n$ steps. _

We define the _first-ladder time_ in $(0,\infty)$ of $S$ (or first-passage
time of $S$) as

(2.2) $\mathcal{T}:=\min\\{n>0\>:\>S_{n}>0\\}\,$

and the corresponding _first-ladder height (or leapover)_ as the control
process stopped at $\mathcal{T}$, i.e. $S_{\mathcal{T}}$. In the same spirit,
we can define the _first-ladder cost_ as the value of the cost process ${C}$
stopped at $\mathcal{T}$, i.e. ${C}_{\mathcal{T}}$. With the choice (2.1),
$L_{\mathcal{T}}(S)$ is the _first-ladder length_ of $S$, that is the length
of the process $S$ up to its first-passage in $(0,\infty)$.

We now give an overview of our main results, characterizing the law of
$C_{\mathcal{T}}$ under different assumptions. The first result is an explicit
expression for the joint generating function of
$(\mathcal{T},S_{\mathcal{T}},{C}_{\mathcal{T}})$ under the assumption that
the joint process $(S,{C})$ has i.i.d. increments. This result will be
instrumental for the analysis of first-ladder quantities related to the random
walk in random media $Y$, though not directly applicable to it. We then
introduce a general process, called _Random Walk in Random Scenery on Bonds_
from its analogy to Random Walks in Random Scenery [17], so to obtain a
suitable representation of $Y_{\mathcal{T}}$ and of $L_{\mathcal{T}}(Y)$ in
this setting. This new process, for which we will state our main result, can
be seen as a cost process coupled with $S$, having dependent increments also
depending on a random scenery assigned to the bonds of ${\mathbb{Z}}$.
Finally, as explicit applications of the main result, we derive the law of the
first-ladder quantities for the random walks in Lévy random media.

###  2.1. Cost process with i.i.d. increments

The investigation of first-ladder time and height of a 1D random walk is
nowadays a well-established topic in fluctuation theory. Among well-known
results, that are derived under the assumption of independent and identically
distributed increments of the walk $S$, the Spitzer-Baxter identity provides
an explicit formula for the generating function of the first-ladder time
$\mathcal{T}$ and height $S_{\mathcal{T}}$:

(2.3)
$\displaystyle\mathbb{E}\left[z^{\mathcal{T}}e^{itS_{\mathcal{T}}}\right]$
$\displaystyle=$ $\displaystyle
1-\exp\left(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\left\\{S_{n}>0\right\\}}e^{itS_{n}}d\mathbb{P}\right)\,.$

In the same spirit of these classical results, assuming that the process
$(S,{C})$ is the sum of i.i.d. random variables222It is worth stressing that
we are not assuming that ${\eta}_{i}$ and $\xi_{i}$ are independent.
$(\xi_{k},{\eta}_{k})_{k\in{\mathbb{N}}}$ we derive an identity akin to the
Spitzer-Baxter identity for the joint control and cost processes:

######  Theorem 2.1.

Suppose that the joint process $(S,{C})$ has i.i.d. increments. Then, for any
$t\in{\mathbb{R}}$, $s\in{\mathbb{R}}^{\ell}$ and $z\in(0,1)$,

(2.4)
$\displaystyle\mathbb{E}\left[z^{\mathcal{T}}e^{itS_{\mathcal{T}}}e^{i{s}\cdot{C}_{\mathcal{T}}}\right]$
$\displaystyle=$ $\displaystyle
1-\exp\left(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\left\\{S_{n}>0\right\\}}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}d\mathbb{P}\right)\,,$
(2.5)
$\displaystyle\mathbb{E}\left[\sum_{n=0}^{\mathcal{T}-1}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]$
$\displaystyle=$
$\displaystyle\exp\left(+\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\left\\{S_{n}\leq
0\right\\}}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}d\mathbb{P}\right)\,.$

As in the classical setting, the identities (2.4) and (2.5) allows to
determine the laws of $\mathcal{T},S_{\mathcal{T}},C_{\mathcal{T}}$ from the
knowledge of quantities that do not depend on the first-passage event (right-
hand sides of the identities).

###  2.2. Random Walk in Random Scenery on Bonds

Let $\zeta^{\pm}=(\zeta^{\pm}_{k})_{k\in{\mathbb{Z}}}$ be two sequences of
i.i.d. real-valued random variables defining the random scenery:
$\zeta^{+}_{k}$ and $\zeta^{-}_{k}$ are the scenery values at bond $k$ of
${\mathbb{Z}}$. In what follows, the random walk $S$ is considered to have
i.i.d. symmetric increments $\xi_{k}$ in ${\mathbb{Z}}$, independent of
$\zeta^{\pm}$. We then consider the cost process
$C=(C_{n})_{n\in{\mathbb{N}}_{0}}$, depending on $S$ and $\zeta^{\pm}$, such
that $C_{0}=0$ and, for $n\in{\mathbb{N}}$,

(2.6) $C_{n}:=\sum_{k=1}^{n}\eta_{k}\,,\quad\qquad\mbox{ with
}\quad\eta_{k}=\begin{cases}\displaystyle\sum_{j=S_{k-1}}^{S_{k}-1}{\zeta}^{+}_{j+1}\,,\qquad&\textrm{if}\quad\xi_{k}>0\,,\\\
\\\ \displaystyle 0&\textrm{if}\quad\xi_{k}=0\,,\\\ \\\
\displaystyle\sum_{j=S_{k}}^{S_{k-1}-1}{\zeta}^{-}_{j+1}\,,\qquad&\textrm{if}\quad\xi_{k}<0\,.\end{cases}$

Basically, each $\eta_{k}$ collects all the scenery values corresponding to
the bonds that have been crossed in the corresponding jump $\xi_{k}$ of $S$,
whereas $\zeta^{+}$ determines the weight associated with bond traversed to
the right and ${\zeta}^{-}$ with bond traversed to the left. In particular,
the cost process $C$ depends on both ${\zeta}^{\pm}$ and $S$ and is called
Random Walk in Random Scenery on Bonds (RWRSB).

The presence of the random scenery breaks down the i.i.d. assumption of the
generalized Spitzer-Baxter identity stated in Theorem 2.1. We will show how to
leverage the results for first-ladder quantities associated with the control
process $S$ to infer properties on the first-ladder quantity $C_{\mathcal{T}}$
in this setting. As it will be clear in the proof, to characterize
$C_{\mathcal{T}}$ we need to consider also the even part of the scenery random
variables, defined as
$\zeta^{0}_{k}\coloneqq\frac{\zeta^{+}_{k}+\zeta^{-}_{k}}{2},\quad\forall
k\in{\mathbb{Z}}$. This analysis will lead to the following:

######  Theorem 2.2.

Let S be a random walk on ${\mathbb{Z}}$ with i.i.d. symmetric increments in
the basin of attraction of a $\beta$-stable distribution. Consider a RWRSB
with ${\zeta}^{+}_{k}$ and ${\zeta}^{0}_{k}$ that are non-negative (or non-
positive) and belong to the domain of attraction of stable random variables
with index $\gamma_{+}$ and $\gamma_{0}$ respectively. Let $C$ be the RWRSB
satisfying (2.6) and set
${\rho}\coloneqq\min\\{\hat{\gamma}_{+}\beta,\,\hat{\gamma}_{0}\hat{\beta}\\}$.
Then, if $\hat{{\gamma}}_{+}{\beta}<\hat{{\gamma}}_{0}\hat{{\beta}}$ , or if
$\,\hat{{\gamma}}_{+}{\beta}>\hat{{\gamma}}_{0}\hat{{\beta}}$ and
$\hat{{\gamma}}_{0}=1$,

(2.7) ${\mathbb{P}}[C_{\mathcal{T}}>x]\sim K(x)\,x^{-{\rho}/2},\\\
\qquad\mbox{as }x\to\infty\,\,,$

where $K(x)$ is a slowly varying function.

Moreover, in all the other cases, the tail distribution of $C_{\mathcal{T}}$
satisfies the same upper bound given in (2.7), while the lower bound, under
the further hypothesis of stability of the ${\zeta}_{k}^{0}$’s, decays
polynomially with exponent
$\min\\{\hat{{\gamma}}_{0},\hat{{\gamma}}_{+}{\beta}/2,\hat{{\beta}}/2\\}$.

We anticipate that a slightly more general theorem is valid under suitable
assumptions (see Thm. 3.13): for the ease of the reader we postpone the
technical details and the complete statement to the dedicated section.

It is worth noting that, when ${\mathbb{E}}({\zeta}^{\pm}_{1})<\infty$, the
exponent of the asymptotic tail of the first-ladder cost is ruled solely by
the properties of the underlying random walk $S$, as expected.

###  2.3. First-ladder quantities for the random walks in Lévy random media
$Y$.

Let ${\zeta}:=({\zeta}_{k})_{k\in{\mathbb{Z}}}$ be a sequence of i.i.d.
positive random variables, whose common distribution belongs to the basin of
attraction of a $\gamma$-stable distribution, with $0<{\gamma}\leq 2$,
${\gamma}\neq 1$. The recursive sequence of definitions

(2.8)
${\omega}_{0}:=0\,,\qquad{\omega}_{k}-{\omega}_{k-1}:={\zeta}_{k}\,,\quad\mbox{for
}k\in{\mathbb{Z}}\,,$

determines a marked _point process_
${\omega}:=({\omega}_{k})_{k\in{\mathbb{Z}}}$ on ${\mathbb{R}}$, which we call
the _random medium_. For a fixed ${\omega}$, we define the discrete time
process $Y:=(Y_{n})_{n\in{\mathbb{N}}_{0}}$ setting

(2.9) $Y_{n}\equiv Y_{n}({\omega},S):={\omega}_{S_{n}}\qquad\forall
n\in{\mathbb{N}}_{0}\,.$

In simple terms, $Y$ performs the same jumps as $S$ but on the points of
${\omega}$, thus it is called _random walk on the random medium_.

The presence of the random medium creates a dependence between the increments
of the process and provides a more realistic model of motion in inhomogeneous
media with respect to the classical hypothesis of i.i.d. jumps. However, as
before, the double source of randomness makes the analysis of the model much
harder than the classical independent case, and even standard results of the
classical theory of random walks, such as transience and recurrence, or
central limit theorems, have been only recently obtained under suitable
hypotheses ([6, 4]). On the other hand it is easy to see that the first-ladder
time $\mathcal{T}$ is the same for both $Y$ and the underlying random walk
$S$. Our aim is thus to characterize the asymptotic law of the first-ladder
length $L_{\mathcal{T}}(Y)$ and height $Y_{\mathcal{T}}$. By noting that $Y$
can be seen as a RWRSB driven by $S$ with scenery
${\zeta}^{+}=-{\zeta}^{-}\equiv{\zeta}$, and similarly for
$(L_{n}(Y))_{n\in{\mathbb{N}}_{0}}$ with the choice
${\zeta}^{+}={\zeta}^{-}\equiv{\zeta}$, we get from Theorem 2.2 the following
results:

######  Corollary 2.3.

(2.10) ${\mathbb{P}}(Y_{\mathcal{T}}>x)\sim
Kx^{-\hat{{\gamma}}{\beta}/2}\,,\qquad\mbox{as }x\to\infty\,\,.$

where $K$ is an explicit constant (see Eqs. 3.42, 3.43).

######  Corollary 2.4.

(2.11) $K_{low}(x)\
\,x^{-\min\left\\{1/2,\hat{{\gamma}}{{\beta}}/2\right\\}}\leq{\mathbb{P}}[L_{\mathcal{T}}(Y)>x]\leq
K_{up}(x)\,x^{-\hat{{\gamma}}\hat{{\beta}}/2},$

with $K_{low}(x)$ and $K_{up}(x)$ suitable slowly varying functions that are
matching if ${{\gamma}},{\beta}\in(1,2]$ (see Subsection 3.3.2).

Notice that, except for the case ${\gamma}\in(0,1)$ and ${\beta}\in(1,2]$, the
exponents of the decay for the lower and upper bounds match.

We are also interested in the continuous-time process $X:=(X_{t})_{t\geq 0}$,
whose trajectories interpolate those of the walk $Y$ and have unit speed.
Formally it can be defined as follows: given a realization ${\omega}$ of the
medium and a realization $S$ of the dynamics, we define the sequence of
_collision times_ $T_{n}\equiv T_{n}({\omega},S)$ via

(2.12) $T_{0}:=0\,,\qquad
T_{n}:=\sum_{k=1}^{n}|{\omega}_{S_{k}}-{\omega}_{S_{k-1}}|\,,\quad\mbox{for
}n\geq 1\,.$

Since the length of the $n^{\mathrm{th}}$ jump of the walk is given by
$|{\omega}_{S_{n}}-{\omega}_{S_{n-1}}|$, $T_{n}$ represents the global length
of the trajectory Y up to the $n^{\mathrm{th}}$ collision. In other words,
$T_{n}=L_{n}(Y)$, and it can be seen as a RWRSB (see also [7]). Finally,
$X_{t}\equiv X_{t}({\omega},S)$ is defined by the equations

(2.13) $X_{t}:=Y_{n}+\mathrm{sgn}(\xi_{n+1})(t-T_{n})\,,\quad\mbox{for
}t\in[T_{n},T_{n+1})\,.$

The process $X$ is also important from the standpoint of applications as it is
a generalization of the so-called Lévy-Lorentz gas [5], that is obtained under
the further assumption that the underlying random walk is simple and
symmetric.

Functional limit theorems for the processes $Y$ and $X$, with suitable
scaling, have been derived in [6, 7, 27] under different set of hypotheses. In
particular, when ${\gamma}\in(0,1)$ or when the underlying random walk
performs heavy-tailed jumps, the processes $Y$ and $X$ are shown to exhibit an
interesting super-diffusive behavior [7, 27].

Let us define the _first-passage time_ in $(0,\infty)$ by

(2.14) $\mathcal{T}(X):=\inf\\{t>0\,:\,X_{t}>0\\}\,.$

Notice that in this continuous setting the notion of first-ladder height
becomes trivial, while that of first-ladder length of $X$ indeed corresponds
to $\mathcal{T}(X)$, being the speed of the process $X$ set equal to 1. By
construction, and using the previous notation, it can be seen that

(2.15)
$\begin{split}&\mathcal{T}(X)=\sum_{k=1}^{\mathcal{T}}|Y_{k}-Y_{k-1}|-Y_{\mathcal{T}}=L_{\mathcal{T}}(Y)-Y_{\mathcal{T}}\,.\end{split}$

This relation shows that, beyond their intrinsic interest, the derivation of
the law of the first-ladder height and length of $Y$ will allow to infer
information on the first-passage time of the process $X$. Indeed, the
continuous first-passage time $L_{\mathcal{T}}(Y)-Y_{\mathcal{T}}$ can be seen
as the value of the RWRSB with scenery on bonds ${\zeta}^{+}\equiv 0\,$ and
${\zeta}^{-}=2{\zeta}$ at time $\mathcal{T}$. As a consequence, we have

######  Corollary 2.5.

(2.16) $\displaystyle K_{low}(x)\
\,t^{-\min\left\\{\hat{{\gamma}},\hat{{\beta}}/2\right\\}}\leq{\mathbb{P}}[{\mathcal{T}}(X)>t]\leq
K_{up}(x)\,t^{-\hat{{\gamma}}\hat{{\beta}}/2},$

where $K_{low}(x)$ and $K_{up}(x)$ are slowly varying functions that are
matching if ${\gamma}\in(1,2]$ (see Subsection 3.3.3).

##  3\. Proofs of results

We give the proof of the main results described in the previous section,
together with some useful corollaries. We also discuss some applications.

###  3.1. Results in the case of $(S,C)$ with i.i.d. increments

In this section we assume that the process $(S,C)$ has i.i.d. increments and
we introduce the characteristic functions

(3.1)
$\phi_{\xi_{1},\eta_{1}}(t,{s}):=\mathbb{E}\left[e^{i(t\xi_{1}+{s}\cdot{\eta_{1}})}\right]\,,\qquad\phi_{{\eta_{1}}}({s}):=\phi_{\xi_{1},{\eta_{1}}}(0,{s})\qquad\mbox{with
}t\in{\mathbb{R}},\,s\in{\mathbb{R}}^{\ell}\,.$

We start proving the generalized Spitzer-Baxter identities stated in Thm. 2.1.
The proof follows the line of that for the classical Spitzer-Baxter identity
as in [10], Paragraph 8.4.

_Proof of Thm. 2.1:_ As $(\xi_{k},{\eta}_{k})_{k\in{\mathbb{N}}}$ are i.i.d.
random variables, we have

(3.2)
$\mathbb{E}\left[\sum_{n=0}^{\infty}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]=\frac{1}{1-z\phi_{\xi_{1},{\eta_{1}}}(t,{s})}=f^{-1}_{+}(z,t,s)f_{-}(z,t,\underline{s})$

where

(3.3a) $\displaystyle f_{+}(z,t,s)$
$\displaystyle:=\exp\left(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\\{S_{n}>0\\}}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}d\mathbb{P}\right),$
(3.3b) $\displaystyle f_{-}(z,t,s)$
$\displaystyle:=\exp\left(+\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\\{S_{n}\leq
0\\}}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}d\mathbb{P}\right).$

Split the sum in the left-hand side of (3.2) as

(3.4)
$\displaystyle\mathbb{E}\left[\sum_{n=0}^{\infty}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{n=0}^{\mathcal{T}-1}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]+\mathbb{E}\left[\sum_{n=\mathcal{T}}^{\infty}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]\,.$

The second term on the right-hand side of (3.4) can be rewritten as

(3.5)
$\displaystyle\mathbb{E}\left[\sum_{n=\mathcal{T}}^{\infty}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[z^{\mathcal{T}}e^{itS_{\mathcal{T}}}e^{i{s}\cdot{C}_{\mathcal{T}}}\sum_{n=0}^{\infty}z^{n}e^{it(S_{n+\mathcal{T}}-S_{\mathcal{T}})}e^{i{s}\cdot({C}_{n+\mathcal{T}}-{C}_{\mathcal{T}})}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[z^{\mathcal{T}}e^{itS_{\mathcal{T}}}e^{i{s}\cdot{C}_{\tau}}\right]/\left(1-z\phi_{\xi_{1},{\eta_{1}}}(t,{s})\right)\,,$

where in the last passage we use the fact that ${S}_{n+\tau}-{S}_{\tau}$ and
${S}_{n}$ have the same distribution, independent of ${S}_{\tau}$, and the
equivalent property for ${C}$. By using (3.2) and (3.5) in (3.4) we get

(3.6)
$\displaystyle\left[1-\mathbb{E}\left[z^{\mathcal{T}}e^{itS_{\mathcal{T}}}e^{i{s}\cdot{C}_{\mathcal{T}}}\right]\right]f_{-}(z,t,s)=f_{+}(z,t,s)\,\mathbb{E}\left[\sum_{n=0}^{\mathcal{T}-1}z^{n}e^{itS_{n}}e^{i{s}\cdot{C}_{n}}\right]\,.$

We now apply standard Wiener-Hopf argument: the convolution of two measures
restricted to $(0,+\infty)$ remains restricted to $(0,+\infty)$ (and the same
for $(-\infty,0]$); by expanding the exponential functions in (3.3), we can
associate $f_{+}$ and $f_{-}$ with $P^{*}$ and $Q^{*}$ in Lemma A.1
respectively, and similarly the remaining terms on both sides of (3.6)
correspond to $P$ and $Q$. The results (2.4) and (2.5) immediately follow
using the lemma. ∎

Theorem 2.1 is particularly useful when the right-hand side of the identities
(2.4) and (2.5) can be computed explicitly. This happens, for example, when
the law of the joint process $(S,C)$ satisfies some symmetry property. In
particular the following definitions will be helpful.

######  Definition 3.1.

The joint process $(S,C)$ is called _s -symmetric_ if, for all
$x\in{\mathbb{R}}$, ${y}\in{\mathbb{R}}^{\ell}$ and $n\in{\mathbb{N}}$,

(3.7) $\,\mathbb{P}(S_{n}\in dx,{C}_{n}\in d{y})=\mathbb{P}(-S_{n}\in
dx,{C}_{n}\in d{y})\,.$

The joint process $(S,C)$ is called _o -symmetric_ if, for all
$x\in{\mathbb{R}}$, ${y}\in{\mathbb{R}}^{\ell}$ and $n\in{\mathbb{N}}$,

(3.8) $\mathbb{P}(S_{n}\in dx,{C}_{n}\in d{y})=\mathbb{P}(-S_{n}\in
dx,-{C}_{n}\in d{y})\,.$

To simplify notations we also define the function

(3.9)
${\Phi}(z,{s}):=\exp\left(-\frac{1}{2\pi}\int_{0}^{\pi}\ln[1-z\phi_{\xi_{1},{\eta_{1}}}(t,{s})]dt\right).$

It is easy to see that the generalized Spitzer-Baxter can be directly used to
give the following explicit relation involving both s-symmetric and
o-symmetric processes, with i.i.d. increments.

######  Corollary 3.2.

Consider a cost process $C^{{\text{\wedn{s}}}}$ such that
$(S,{C}^{\text{\wedn{s}}})$ is s-symmetric, together with a cost process
${C}^{{\text{\wedn{o}}}}$ such that $(S,C^{{\text{\wedn{o}}}})$ is
o-symmetric, all with i.i.d. increments. Then, for all $z\in(0,1)$ and
$s\in{\mathbb{R}}^{\ell}$,

(3.10)
$\left(1-\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot({C}_{\mathcal{T}}^{{\text{\wedn{s}}}}+{C}_{\mathcal{T}}^{{\text{\wedn{o}}}})}\right]\right)\left(1-\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot({C}_{\mathcal{T}}^{{\text{\wedn{s}}}}-{C}_{\mathcal{T}}^{{\text{\wedn{o}}}})}\right]\right)=\left(1-z\phi_{{\eta_{1}}}({s})\right){\Phi}^{2}(z,{s})\,,$

where $\eta=(\eta_{k})_{k\in{\mathbb{N}}}$ are the cost increments associated
with ${C}^{{\text{\wedn{s}}}}+{C}^{{\text{\wedn{o}}}}$.

###### Proof.

Choosing $C=C^{{\text{\wedn{s}}}}\pm{C}^{{\text{\wedn{o}}}}$ in (2.4), we get
for $t=0$

(3.11) $\displaystyle
1-\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot({C}_{\mathcal{T}}^{{\text{\wedn{s}}}}\pm{C}_{\mathcal{T}}^{{\text{\wedn{o}}}})}\right]$
$\displaystyle=$
$\displaystyle\exp\left(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\left\\{S_{n}>0\right\\}}e^{i{s}\cdot({C}_{n}^{{\text{\wedn{s}}}}\pm{C}_{n}^{{\text{\wedn{o}}}})}d\mathbb{P}\right),$

On the other hand, using the definitions (3.7) and (3.8), we also have

(3.12)
$\int_{\left\\{S_{n}>0\right\\}}e^{i{s}\cdot{({C}_{n}^{{\text{\wedn{s}}}}-{C}_{n}^{{\text{\wedn{o}}}})}}\,\,d\mathbb{P}=\int_{\left\\{S_{n}<0\right\\}}e^{i{s}\cdot{({C}_{n}^{{\text{\wedn{s}}}}+{C}_{n}^{{\text{\wedn{o}}}})}}\,\,d\mathbb{P}\,.$

Putting together Eqs. (3.11) and (3.12), and using the i.i.d. assumption about
the increments $\eta$ of ${C}^{{\text{\wedn{s}}}}+{C}^{{\text{\wedn{o}}}}$, we
have

$\displaystyle\left(1-\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot({C}_{\mathcal{T}}^{{\text{\wedn{s}}}}+{C}_{\mathcal{T}}^{{\text{\wedn{o}}}})}\right]\right)\left(1-\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot({C}_{\mathcal{T}}^{{\text{\wedn{s}}}}-{C}_{\mathcal{T}}^{{\text{\wedn{o}}}})}\right]\right)$
$\displaystyle=$
$\displaystyle\frac{(1-z\phi_{{\eta_{1}}}({s}))}{\exp\left(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\left\\{S_{n}=0\right\\}}e^{i{s}\cdot{({C}_{n}^{{\text{\wedn{s}}}}+{C}_{n}^{{\text{\wedn{o}}}})}}d\mathbb{P}\right)}\,.$

The integral at denominator in the r.h.s. of the above equation is the anti-
transform with respect to $S_{n}$, evaluated at $S_{n}=0$, of the joint
transform of a $n^{\mathrm{th}}$ convolution of $(\xi,{\eta})$. Hence, by
using again the i.i.d. assumption about the increments
$(\xi_{k},{\eta}_{k})_{k\in{\mathbb{N}}}$, and the fact that the convolution
becomes a product in the transform domain, we get

(3.13)
$\displaystyle\exp\left(+\sum_{n=1}^{\infty}\frac{z^{n}}{n}\int_{\left\\{S_{n}=0\right\\}}e^{i{s}\cdot{({C}_{n}^{{\text{\wedn{s}}}}+{C}_{n}^{{\text{\wedn{o}}}})}}d\mathbb{P}\right)$
$\displaystyle=$
$\displaystyle\exp\left(+\sum_{n=1}^{\infty}\frac{z^{n}}{n}\frac{1}{\pi}\int_{0}^{\pi}\phi_{\xi_{1},\eta_{1}}(t,{s})^{n}dt\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{\pi}\int_{0}^{\pi}\ln[1-z\phi_{\xi_{1},\eta_{1}}(t,{s})]dt\right)$
$\displaystyle=$ $\displaystyle{\Phi}(z,s)$

from which the result follows. ∎

In the s-symmetric case, Eq. (3.10) allows to obtain an explicit
representation of the generating function of the first-ladder cost:

######  Corollary 3.3.

If $(S,C)$ is _s -symmetric_ with i.i.d. increments, then

(3.14)
$\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot{C}_{\mathcal{T}}}\right]=1-\sqrt{1-z\phi_{{\eta_{1}}}({s})}{\Phi}(z,{s})\,.$

###### Proof.

Directly follows form (3.10) with $C^{{\text{\wedn{o}}}}\equiv 0$. ∎

######  Remark 3.4.

If $(S,C)$ is s-symmetric with i.i.d. increments, and the control process $S$
has continuous distribution, then ${\Phi}(z,s)=1$ and thus

(3.15)
$\mathbb{E}\left[z^{\mathcal{T}}e^{i{s}\cdot{C}_{\mathcal{T}}}\right]=1-\sqrt{1-z\phi_{{\eta_{1}}}({s})}\,,$

leading to the behavior of ${C}_{\mathcal{T}}$ stated in [2], where a
combinatorial proof of this result has been provided. Notice that the
dependence of this joint generating function on the random walk distribution
comes only through the costs, that in general depend on $S$. The presence of a
discrete jump distribution yields a correction term ${\Phi}(z,{s})$ that
instead explicitly depends on the random walk, as already underlined in [19].
Observe also that the identity (3.15) generalizes the classical Sparre-
Andersen identity, which is recovered for ${s}={0}$.

The generalized Spitzer-Baxter identities stated in Theorem 2.1, together with
Corollaries 3.2 and 3.3, provide the key element to identify the law of the
first-ladder quantities involved in it (see e.g. [10] for the classical
treatment).

While the laws of $\mathcal{T}$ and $S_{\mathcal{T}}$ are well known under
quite general hypotheses on the random walk $S$ with i.i.d. increments (see
[12] and references therein), the focus will rather be given to the first-
ladder cost. We now derive the asymptotic distribution of $C_{\mathcal{T}}$,
under the assumption that the joint process is s-symmetric or o-symmetric.

######  Proposition 3.5.

Assume that $(S,C)$ is s-symmetric with i.i.d. increments
$(\xi_{k},\eta_{k})_{k\in{\mathbb{N}}}$. If $\eta_{1}$ is in the basin of
attraction of a $\gamma$-stable law, then $C_{\mathcal{T}}$ is in the basin of
attraction of a $\hat{{\gamma}}/2$-stable law. More precisely

  * (A)

If $\phi_{\eta_{1}}(s)=1+i{\nu}s+o(s)$ for $s\to 0^{+}$, with ${\nu}>0$ real
and finite (similarly if ${\nu}<0$), then as $x\to+\infty$

(3.16)
$\displaystyle{\mathbb{P}}[C_{\mathcal{T}}>x]\sim\sqrt{\frac{{\nu}}{\pi}}{\Phi}(1,0)x^{-1/2}\,,\quad{\mathbb{P}}[C_{\mathcal{T}}<-x]=o(x^{-1/2})\,,$

  * (B)

If $\phi_{\eta_{1}}(s)=1-c_{1}s^{{\gamma}}+o(s^{{\gamma}})$, for $s\to 0^{+}$,
${\gamma}\in(0,1]$ and $c_{1}\in\mathbbm{C}$ a complex constant with
$\Re(c_{1})>0$, then as $x\to\infty$

(3.17)
${\mathbb{P}}[C_{\mathcal{T}}>x]\sim\frac{Cp_{+}}{{\Gamma}\left(1-{\gamma}/2\right)}\,x^{-{\gamma}/2}\,,\quad{\mathbb{P}}[C_{\mathcal{T}}<-x]\sim\frac{Cp_{-}}{{\Gamma}\left(1-{\gamma}/2\right)}\,x^{-{\gamma}/2}\,,$

where

$C=\frac{{\Phi}(1,0)}{\cos(\pi{\gamma}/4)}[\Re(c_{1})^{2}+\Im(c_{1})^{2}]^{1/4}\cos\left(\frac{1}{2}\arctan\left(\frac{\Im(c_{1})}{\Re(c_{1})}\right)\right)\,,$

and

$p_{+}=1-p_{-}=\frac{1}{2}\left(1-\frac{\sin\left(\frac{1}{2}\arctan\left(\frac{\Im(c_{1})}{\Re(c_{1})}\right)\right)}{\cos\left(\frac{1}{2}\arctan\left(\frac{\Im(c_{1})}{\Re(c_{1})}\right)\right)\tan\left(\frac{\pi{\gamma}}{4}\right)}\right)\in[0,1]\,.$

If $p_{+}=0$ or $p_{-}=0$, then we interpret (3.17) as $o(x^{-{\gamma}/2})$.

  * (C)

If $\phi_{\eta_{1}}(s)=1+ic_{2}s\log(1/s)+o(s\log(1/s))$, with $s\to 0^{+}$
and $c_{2}\in\mathbbm{R}$ a positive constant (similarly for $c_{2}<0$), then
as $x\to\infty$

(3.18)
$\displaystyle{\mathbb{P}}[C_{\mathcal{T}}>x]\sim\sqrt{\frac{c_{2}\log(x)}{\pi}}{\Phi}(1,0)x^{-1/2}\,,\quad{\mathbb{P}}(C_{\mathcal{T}}(Z)<-x)=o\left(\frac{\sqrt{\log(x)}}{x^{1/2}}\right)\,,$

  * (D)

If $\phi_{\eta_{1}}(s)=1-c_{3}s^{{\gamma}}+o(s^{{\gamma}})$, with $s\to
0^{+}$, ${\gamma}\in(1,2]$ and $c_{3}\in\mathbbm{R}^{+}$ a positive constant,
then as $x\to\infty$

(3.19)
${\mathbb{P}}[C_{\mathcal{T}}>+x]\sim{\mathbb{P}}[C_{\mathcal{T}}<-x]\sim\frac{C}{2{\Gamma}\left(1-{\gamma}/2\right)}\,x^{-{\gamma}/2}\,,$

where

$C={\Phi}(1,0)\sqrt{c_{3}}\begin{cases}\displaystyle
1/\cos(\pi{\gamma}/4)\,,\qquad&\mbox{ if }\quad{\gamma}\in(1,2)\,,\\\
\displaystyle 2/\pi\,,\qquad&\mbox{ if }\quad{\gamma}=2\,.\end{cases}$

###### Proof.

From Corollary 3.3, we have

${\mathbb{E}}\left[e^{isC_{\mathcal{T}}}\right]=1-\sqrt{1-\phi_{\eta_{1}}(s)}{\Phi}(1,s)\,.$

Since the result depends solely on the behaviour of the characteristic
function $\phi_{\eta_{1}}$ around $0$ (see e.g. [14]), or equivalently on the
tail distributions of $\eta_{1}$, the tail asymptotic of $C_{\mathcal{T}}$ can
be readily determined. ∎

Similarly, in the o-symmetric case we have the following:

######  Proposition 3.6.

Assume that $(S,C)$ is o-symmetric with i.i.d. increments
$(\xi_{k},\eta_{k})_{k\in{\mathbb{N}}}$, and let ${\gamma}\in(0,2]$ such that
$\phi_{\eta_{1}}(s)=1-c_{4}s^{\gamma}+o(s^{\gamma})$ for some
$c_{4}\in\mathbbm{R^{+}}$. In the above notation, and for all ${\gamma}\neq
2$, it holds that

(3.20) ${\mathbb{P}}[|C_{\mathcal{T}}|>x]\sim K\cdot x^{-{\gamma}/2}\,,$

where the constant is explicit
$K={\Phi}(1,0)\sqrt{c_{4}}/{\Gamma}(1-{\gamma}/2)$ whenever $C_{\mathcal{T}}$
is non-negative (or non-positive).

###### Proof.

The proof follows by setting $z=1$ in Eq. (3.10), and performing a series
expansion around $s=0$ on both sides. More specifically, the ansatz $\phi_{\pm
C_{\mathcal{T}}}(s)=1-c_{\pm}s^{\alpha}+o(s^{\alpha})$, with $c_{\pm}$ complex
conjugate constants, provides $|c_{+}|=|c_{-}|=\sqrt{c_{4}}{\Phi}(1,0)$ and
${\alpha}={\gamma}/2$.

If ${\gamma}\in(0,2)$, then ${\alpha}\in(0,1)$ and it turns out that
$\Re(c_{\pm})\neq 0$. Furthermore, for a non-negative cost process we know
that $c_{\pm}=ce^{\mp i\frac{\pi}{2}{\alpha}}\,$, which concludes the proof.
Notice that if ${\gamma}=2$ (and thus ${\alpha}=1$) we do not know if
$\Re(c_{\pm})\neq 0$, and hence we can not draw any conclusions about the tail
distribution of $C_{\mathcal{T}}$. ∎

####  3.1.1. Applications

The s-symmetric (o–symmetric) condition is fulfilled in the following
situations. Consider a joint process $(S,C)$ with i.i.d. increments
$(\xi_{k},\eta_{k})_{k\in{\mathbb{N}}}$ such that, for a given function
$g:{\mathbb{R}}\mapsto{\mathbb{R}}^{\ell}$,

(3.21) $\eta_{k}={g}(\xi_{k})\qquad\forall k\in{\mathbb{N}}.$

It is apparent that if the function $g$ is even (odd) the joint process
$(S,C)$ is s-symmetric (o-symmetric). As a main example, let us consider the
one-dimensional cost process $C\equiv L$ defined in (2.1), corresponding to
the length of the process $S$, obtained by choosing $g(\xi_{k})=|\xi_{k}|$.
Applying the above result we will obtain a complete characterization of the
asymptotic law of the first-ladder length $L_{\mathcal{T}}(S)$. Similarly, by
choosing $g(\xi_{k})=\xi_{k}$ we will fully characterize the asymptotic
behavior of the first-ladder height (or leapover) $S_{\mathcal{T}}$. Both
results will be of great use in the next section, we thus state them
explicitly for the ease of later reference.

######  Corollary 3.7.

Let S have i.i.d. symmetric increments in the domain of attraction of a
$\beta$-stable law. Then the first-ladder length $L_{\mathcal{T}}(S)$ is in
the domain of attraction of a ${\hat{{\beta}}}/2$-stable law. More precisely,
writing $\phi_{\xi_{1}}(s)=1-\nu s^{{\beta}}+o(s^{{\beta}})$, we have

(3.22)
${\mathbb{P}}(L_{\mathcal{T}}(S)>x)\sim\frac{\sqrt{C}}{{\Gamma}\left(1-\hat{{\beta}}/2\right)}{\Phi}(1,0)x^{-\hat{{\beta}}/2}\,\qquad\mbox{as
}x\to\infty\,,$

where

$C\coloneqq\begin{cases}{\nu}/\cos(\pi\hat{{\beta}}/2)\qquad&\mbox{if}\quad{\beta}\in(0,1)\\\
2{\nu}/\pi\log(x)\qquad&\mbox{if}\quad{\beta}=1\\\
{\mathbb{E}}[|\xi_{1}|]\qquad&\mbox{if}\quad{\beta}\in(1,2]\end{cases}\,,$

######  Remark 3.8.

As a notable application, consider the situation in which a random time is
needed to perform a jump for the random walker $S$. In this case, the total
time can be considered (per our notation) as a cost associated with the random
walk. In particular (but see [2] for details and physical motivations) suppose
that the time taken to perform a jump is correlated with its length. This is
indeed the case for 1D Lévy walks, which are a continuous-time interpolation
(with unit speed) of 1D $RW$ with i.i.d. and heavy-tailed jumps, a.k.a. Lévy
flights [28]. Thus, $L_{\mathcal{T}}(S)$ corresponds to the first-passage time
for the wait-then-jump model associated with a Lévy walk, as mentioned in [2].

######  Remark 3.9.

It is worthwhile to point out that Corollary 3.7 extends and completes a
previous result by Sinai (see [21, Theorem 3]). One can easily retrace his
proof in the presence of an appropriate cost, still fulfilling necessary
hypotheses, in order to get the basin of attraction of $L_{\mathcal{T}}(S)$
rather than the leapover, but under the assumption that the random variables
$\xi_{k}$’s have stable distribution.

The domain of attraction of the leapover $S_{\mathcal{T}}$, instead, stems
from standard results of fluctuation theory (see [12] and references therein).
Here is obtained by applying Proposition 3.6:

######  Corollary 3.10.

Let S have i.i.d. symmetric increments in the domain of attraction of a
$\beta$-stable law with $\phi_{\xi_{1}}(s)=1-\nu s^{{\beta}}+o(s^{{\beta}})$.
Then the first-ladder length $S_{\mathcal{T}}(S)$ is in the domain of
attraction of a ${{{\beta}}}/2$-stable law. More precisely

(3.23)
${\mathbb{P}}(S_{\mathcal{T}}>x)\sim\frac{\sqrt{{\nu}}}{{\Gamma}\left(1-{\beta}/2\right)}{\Phi}(1,0)x^{-{\beta}/2}\,\qquad\mbox{as
}x\to+\infty\,.$

Notice that the limiting case ${\beta}=2$ is included in [12, Theorem 4],
hence the tail asymptotic ${\mathbb{P}}(S_{\mathcal{T}}>x)\propto x^{-1}$
still holds true.

Another helpful tool, that will be used repeatedly throughout the proof of our
main result, concerns linear combinations of the random variables
$L_{\mathcal{T}}(S)$ and $S_{\mathcal{T}}$:

######  Lemma 3.11.

Consider a cost process $C_{\mathcal{T}}$ with characteristic function

$\mathbb{E}\left[e^{isC_{\mathcal{T}}}\right]=\mathbb{E}\left[e^{ia(s+o(s))L_{\mathcal{T}}(S)}\cdot
e^{ib(s+o(s))S_{\mathcal{T}}}\right]\,,$

where $a,\,b$ are given constants satisfying $a,\,b\in\mathbbm{R}$, $a\neq 0$.
The basin of attraction of the random variable $C_{\mathcal{T}}(S)$ is the
same as for $L_{\mathcal{T}}(S)$.

###### Proof.

The proof has to be split according to the fact that $b$ is of the same sign
as $a$ or of opposite sign. Without loss of generality, we can assume that
$a>0$: when $a<0$, inequalities are intended to be reversed. Firstly, note
that the case $b=0$ is trivial. If $b\neq 0$ we have the following cases.

  1. (i)

If $b>0$, it is enough to observe that for sufficiently small $s$

$0<(a+o(1))L_{\mathcal{T}}(S)\leq
C_{\mathcal{T}}(S)\leq(a+b+o(1))L_{\mathcal{T}}(S),$

given that $0<S_{\mathcal{T}}\leq L_{\mathcal{T}}(S)$. In fact, by applying
Corollary 3.7, we can immediately conclude that
${\mathbb{P}}[C_{\mathcal{T}}(S)>x]\propto x^{-\hat{\beta}/2}$.

  2. (ii)

If $b<0$, as long as $a+b>0$ the above argument is still valid by reversing
the inequality sign. When $a+b\leq 0$, instead, we have to exploit (i)
together with Corollary 3.2. More specifically, we can write

(3.24) $\left(1-\mathbb{E}\left[e^{is(aL_{\mathcal{T}}\pm
bS_{\mathcal{T}})}\right]\right)\left(1-\mathbb{E}\left[e^{is(aL_{\mathcal{T}}\mp
bS_{\mathcal{T}})}\right]\right)=\left(1-{\mathbb{E}}[e^{is(a|\xi_{1}|\pm
b\xi_{1})}]\right){\Phi}^{2}(1,s)\,,$

where we have omitted terms of order $o(s)$ to lighten the notation. Notice
that higher order corrections are irrelevant to the final result, since we are
only interested in the leading term of the asymptotic expansion. It is obvious
that the right-hand side of Eq. (3.24) is proportional to $s^{\hat{\beta}}$,
since ${\mathbb{E}}[a|\xi_{1}|\pm b\xi_{1}]=a{\mathbb{E}}[|\xi_{1}|]\neq 0$,
and $1-\phi_{aL_{\mathcal{T}}-bS_{\mathcal{T}}}(s)\propto s^{\hat{\beta}/2}$
thanks to (i). The desired conclusion immediately follows by comparing both
sides of Eq. (3.24).

∎

It is worth noting that, as long as ${\beta}\in[1,2]$, or if ${\beta}\in(0,1)$
and $|a|^{\hat{\beta}}\neq|b|^{\hat{\beta}}\cos(\pi\hat{\beta}/2)$, the
statement can be seen alternatively as a direct consequence of Lemma B.1.

######  Remark 3.12.

In the presence of spatio-temporal correlations, as explained in Remark 3.8,
notice that the cost process $C_{\mathcal{T}}$ defined in Lemma 3.11 with
$b=-a$ corresponds to the first-passage time
$L_{\mathcal{T}}(S)-S_{\mathcal{T}}$ for the Lévy Walk.

###  3.2. Results for ladder costs associated with RWRSB

The focus of the present section is the cost process $C=C(S,{\zeta}^{\pm})$
defined in Section 2, and called RWRSB. We remind that the process $C$
collects all the scenery values $\zeta^{\pm}_{k}$ corresponding to the bonds
that have been crossed in every jump of $S$, taking into account also the
travel direction. In particular, the random scenery creates a dependence
between the increments of $C$, and breaks down the i.i.d. assumption of the
generalized Spitzer-Baxter identity stated in Theorem 2.1.

In this subsection, we will study the first-ladder costs associated with
RWRSB, and extend the results derived in the previous subsection to this
general context. This analysis will lead to Theorem 2.2, that provides the
asymptotic distribution of $C_{\mathcal{T}}$ under the assumption that the
underlying random walk has i.i.d. symmetric increments. As stressed just after
Theorem 2.2, our main result is now stated and proved emphasizing all the
different scenarios arising as the parameters of the problem vary.

As a final observation, we underline that the extension to ladder costs
$(C_{\mathcal{T}_{k}})_{k\in\mathbbm{N}_{0}}$, where $\mathcal{T}_{k}$ is the
ladder time corresponding to the $k$-th maximum value reached by $Y$, will be
directly dealt with along the proof of the main theorem.

First of all, let us explicitly fix some notations. We consider a symmetric
underlying random walk $S$ in ${\mathbb{Z}}$ with i.i.d. discrete increments
$(\xi_{k})_{k\in\mathbbm{N}}$, whose corresponding characteristic function is,
for $s\to 0^{+}$,

(3.25) $\phi_{\xi_{1}}(s)=1-\nu s^{{\beta}}+o(s^{{\beta}})$

with ${\beta}\in(0,2]$ and ${\nu}\in\mathbbm{R}^{+}$. The related
characteristic function of $|\xi_{1}|$ is, for $s\to 0^{+}$, of the form

(3.26)
$\phi_{|\xi_{1}|}(s)=1+\hat{{\nu}}s^{\hat{{\beta}}}+o(s^{\hat{{\beta}}})\,,$

where

$\hat{{\nu}}=\begin{cases}-{\nu}[1-i\tan(\pi{\beta}/2)]\,,\qquad&\mbox{for}\quad{\beta}\in(0,1)\,,\\\
-{\nu}[1-i\frac{2}{\pi}\log(1/s)]\,,\qquad&\mbox{for}\quad{\beta}=1\,,\\\
i{\mathbb{E}}[|\xi_{1}|]\,,\qquad&\mbox{for}\quad{\beta}\in(1,2]\,.\end{cases}$

We also explicitly write the common characteristic function of the random
variables $\zeta_{k}^{\pm}$’s that we suppose being in the basin of attraction
of a ${\gamma}^{\pm}$-stable law respectively, with
${\gamma}^{\pm}\in(0,2],\,{\gamma}^{\pm}\neq 1$: for ${\theta}\to 0^{+}$

(3.27) $\displaystyle\phi_{{\zeta}_{1}^{\pm}}({\theta})$
$\displaystyle=\begin{cases}1-c_{\pm}{\theta}^{{\gamma}_{\pm}}+o({\theta}^{{\gamma}_{\pm}})\,,\quad&{\gamma}_{\pm}=\hat{{\gamma}}_{\pm}\in(0,1)\,;\>c_{\pm}\in\mathbbm{C}\,,\>\Re(c_{\pm})>0\,,\\\
1+i{\mu}_{\pm}{\theta}-c_{\pm}{\theta}^{{\gamma}_{\pm}}+o({\theta}^{{\gamma}_{\pm}})\,,\quad&{{\gamma}}_{\pm}\in(1,2]\,,\hat{{\gamma}}_{\pm}=1\,;\>{\mu}_{\pm}\in\mathbbm{R}\,,\>c_{\pm}\in\mathbbm{C}\,,\\\
1\,,\quad&{\gamma}_{\pm}=\hat{{\gamma}}_{\pm}=+\infty\;\implies\;{\zeta}_{1}^{\pm}\equiv
0\,.\end{cases}$

We will also need to refer to the even part of the scenery values
$\zeta_{k}^{0}=\frac{\zeta_{k}^{+}+\zeta_{k}^{-}}{2}$, for all
$k\in{\mathbb{Z}}$, and assume that their common characteristic function is
given by

(3.28) $\displaystyle\phi_{{\zeta}_{1}^{0}}({\theta})$
$\displaystyle=\begin{cases}1-c_{0}{\theta}^{{\gamma}_{0}}+o({\theta}^{{\gamma}_{0}})\,,\quad&{\gamma}_{0}=\hat{{\gamma}}_{0}\in(0,1)\,;\>c_{0}\in\mathbbm{C}\,,\>\Re(c_{0})>0\,,\\\
1+i{\mu}_{0}{\theta}-c_{0}{\theta}^{{\gamma}_{0}}+o({\theta}^{{\gamma}_{0}})\,,\quad&{{\gamma}}_{0}\in(1,2]\,,\hat{{\gamma}}_{0}=1\,;\;\>{\mu}_{0}\in\mathbbm{R}\,,\>c_{0}\in\mathbbm{C}\,,\\\
1\,,\quad&{\gamma}_{0}=\hat{{\gamma}}_{0}=+\infty\;\implies\;{\zeta}_{1}^{0}\equiv
0\,.\end{cases}$

Sometimes we will ask for the extra-assumption that the $\zeta_{k}^{0}$’s have
stable distribution with index ${\gamma}_{0}$. Along the proof, we will
explicitly mention where and when this hypothesis enters in a fundamental way.

Let us discuss the relationship between the cost exponents
$\hat{{\gamma}}_{\pm}$ and $\hat{{\gamma}}_{0}$, which will be crucial for the
structure of the proof. By applying Lemma B.1, it is easy to verify that

  * •

if $\hat{{\gamma}}_{+}\neq\hat{{\gamma}}_{-}$, we have
$\hat{{\gamma}}_{0}=\min\\{\hat{{\gamma}}_{+},\hat{{\gamma}}_{-}\\}$

  * •

if $\hat{{\gamma}}_{+}=\hat{{\gamma}}_{-}$:

    * –

$\hat{{\gamma}}_{0}=\hat{{\gamma}}_{+}=\hat{{\gamma}}_{-}$

    * –

$\hat{{\gamma}}_{0}>\hat{{\gamma}}_{+}=\hat{{\gamma}}_{-}\,$, including two
possible cases:

      1. (a)

$(0,1]\ni\hat{{\gamma}}_{0}>\hat{{\gamma}}_{+}=\hat{{\gamma}}_{-}\in(0,1)\,\implies$
${\zeta}_{1}=-{\zeta}_{1}+h({\zeta}_{1})$ with $h({\zeta}_{1})\neq 0$ and
$\hat{{\gamma}}_{0}\equiv\hat{{\gamma}}_{h({\zeta}_{1})}\,$;

      2. (b)

$\hat{{\gamma}}_{0}=+\infty\implies{\zeta}_{1}^{0}\equiv 0\,.$

Finally, let $(\mathcal{T}_{n})_{n\geq 0}$ be the sequence of ladder times of
the process $Y$, namely the consecutive times when the random walk attains a
new maximum value. Formally, they are recursively defined by

$\mathcal{T}_{0}=0\,,\quad\mathcal{T}_{n}:=\min\\{k>\mathcal{T}_{n-1}\,:\,Y_{k}>Y_{\mathcal{T}_{n-1}}\\}\quad\forall
n\in{\mathbb{N}}\,,$

so that $\mathcal{T}_{1}\equiv\mathcal{T}$. Notice that by construction they
are equivalent to the ladder times of the underlying random walk $S$ and in
particular, by the Markov property, they give rise to a renewal process.

To state the main theorem, it is also convenient to define the following
constants:

${\rho}_{+}\coloneqq\hat{{\gamma}}_{+}\,{\beta},\quad\,{\rho}_{0}\coloneqq\hat{{\gamma}}_{0}\,\hat{{\beta}}.$

Note that ${\rho}_{+}$ as well as ${\rho}_{0}$ involves the stability indexes
of both the scenery values and the underlying random walk. Indeed we
heuristically expect that the asymptotic tail of the ladder cost should
receive contributions from both elements of randomness, as the RWRSB
interlaces the two (otherwise independent) processes. The two exponent will
enters in the asymptotic tail of the cost, as substantiated in the proof of
the following theorem.

######  Theorem 3.13.

Let $C$ be a cost process satisfying (2.6), $(C_{\mathcal{T}_{n}})_{n\geq 0}$
the corresponding ladder cost process. Suppose that ${\zeta}^{+}$ and
${\zeta}^{0}$ are i.i.d. sequences of non-negative (similarly for non-
positive) random variables. Then

  * •

If ${\rho}_{+}<{\rho}_{0}\,$, then $\exists K\in\mathbbm{R}^{+}$ explicit
constant such that

${\mathbb{P}}[C_{\mathcal{T}_{n}}>x]\sim K\cdot n\cdot x^{-{\rho}_{+}/2}\,.$

Moreover, if $\hat{{\gamma}}_{0}=+\infty$, we can relax the hypothesis about
the sign of the ${\zeta}_{k}$’s, and refer to
${\mathbb{P}}[|C_{\mathcal{T}_{n}}|>x]$ in the statement.

  * •

If ${\rho}_{+}>{\rho}_{0}\ $, $\hat{{\gamma}}_{0}=1$, then $\exists K(x)$
slowly varying function such that

${\mathbb{P}}[C_{\mathcal{T}_{n}}>x]\sim K(x)\cdot n\cdot
x^{-{\rho}_{0}/2}\,,$

where $K(x)$ is $K\sqrt{\log(x)}$ with $K\in\mathbbm{R}^{+}$ if ${\beta}=1$
and a real positive constant otherwise.

  * •

If ${\rho}_{+}\geq{\rho}_{0}\ $, $\hat{{\gamma}}_{0}\in(0,1)$ and the
${\zeta}_{k}^{0}$’s have stable distribution, or if ${\rho}_{+}={\rho}_{0}\ $
and $\hat{\gamma}_{0}=1$, then $\exists K_{low}(x),K_{up}(x)$ slowly varying
functions such that

$K_{low}(x)\cdot n\cdot
x^{-\min\left\\{\hat{{\gamma}}_{0},\frac{\rho_{+}}{2},\frac{\hat{{\beta}}}{2}\right\\}}\leq{\mathbb{P}}[C_{\mathcal{T}_{n}}>x]\leq
K_{up}(x)\cdot n\cdot x^{-{\rho}_{0}/2}.$

Moreover: The upper bound holds true even without the stability assumption on
${\zeta}^{0}$; If $\hat{{\gamma}}_{+}=+\infty$ we can remove this assumption
also for the lower bound.
Finally, $K_{up}(x)\equiv K_{up}\in\mathbbm{R}^{+}$ and $K_{low}(x)\equiv
K_{low}\in\mathbbm{R}^{+}\,$ unless $\hat{\gamma}_{0}=1$ and ${\beta}=1$, for
which $K_{low}(x)=K_{up}(x)=K\sqrt{\log(x)}$ with $K\in\mathbbm{R}^{+}$, or
$\hat{\gamma}_{0}\in(0,1)$ and ${\beta}=1$, for which
$K_{up}(x)=K_{up}\sqrt{\log(x)}$ with $K_{up}\in\mathbbm{R}^{+}$ and similarly
for $K_{low}(x)$ if the lower tail exponent is given by $\hat{\beta}/2$.

######  Remark 3.14.

We note explicitly that when the bounds on the asymptotic law are not
matching, they do not permit to infer the basin of attraction of the
corresponding quantities. However, we anticipate that while the lower bounds
are crude estimates, the upper bounds need a more refined argument. For this
reason, we expect that the upper bounds provide the correct asymptotic
behavior.

##### Preliminary tools

Following [7], it is convenient to introduce the family of random variables
$\mathcal{N}_{n}(k)$, for $k\in{\mathbb{Z}}$ and $n\in{\mathbb{N}}$, called
_local times on the bonds_ of the random walk $S$, and given by

(3.29)
$\mathcal{N}_{n}(k)\coloneqq\\#\\{j\in\\{1,\ldots,n\\}\,:\,[k-1,k]\subseteq[S_{j-1},S_{j}]\\}\,,$

where the notation $[a,b]$ denotes the closed interval between the real
numbers $a$ and $b$, irrespective of their order. In other words,
$\mathcal{N}_{n}(k)$ is the number of times that the walk $S$ travels the bond
$[k-1,k]$ and in turn can be split into
$\mathcal{N}_{n}(k)=\mathcal{N}_{n}^{-}(k)+\mathcal{N}_{n}^{+}(k)$ where

$\displaystyle\mathcal{N}_{n}^{-}(k)$ $\displaystyle\coloneqq$
$\displaystyle\\#\\{j\in\\{1,\ldots,n-1\\}\,:\,S_{j}\geq k\,,S_{j+1}\leq
k-1\\}\,,$ $\displaystyle\mathcal{N}_{n}^{+}(k)$ $\displaystyle\coloneqq$
$\displaystyle\\#\\{j\in\\{1,\ldots,n-1\\}\,:\,S_{j+1}\geq k\,,S_{j}\leq
k-1\\}\,,$

denote the number of crossings of $[k-1,k]$, respectively, from right to left
and from left to right. In the following, it will be useful to express the
first-ladder height and length of the process $S$ in terms of local times. An
easy check shows that

(3.30a) $\displaystyle\mbox{if}\;k\leq 0\,,\quad$
$\displaystyle\mathcal{N}_{\mathcal{T}}^{+}(k)=\mathcal{N}_{\mathcal{T}}^{-}(k)=\frac{\mathcal{N}_{\mathcal{T}}(k)}{2}\,,\quad$
$\displaystyle\sum_{k\leq
0}\mathcal{N}_{\mathcal{T}}(k)=L_{\mathcal{T}}(S)-S_{\mathcal{T}}\,,$ (3.30b)
$\displaystyle\mbox{if}\;k>0\,,\quad$
$\displaystyle\mathcal{N}_{\mathcal{T}}(k)=\begin{cases}1\;&k\leq
S_{\mathcal{T}}\\\ 0\;&k>S_{\mathcal{T}}\end{cases}\,,\quad$
$\displaystyle\sum_{k>0}\mathcal{N}_{\mathcal{T}}(k)=S_{\mathcal{T}}\,.$

Moreover, since the local times are non-negative integers, we can point out
the following estimates. Without loss of generality, assume that
${\gamma}\in(0,1)$ (when ${\gamma}\in(1,2]$ it is sufficient to reverse the
inequality sign). Then, $\forall\>k\in{\mathbb{Z}}$, we have
$\left(\mathcal{N}_{\mathcal{T}}(k)\right)^{{\gamma}}\leq\mathcal{N}_{\mathcal{T}}(k)$,
and summing over bonds on both sides we get

(3.31) $\sum_{k\leq 0}\left(\mathcal{N}_{\mathcal{T}}(k)\right)^{{\gamma}}\leq
L_{\mathcal{T}}(S)-S_{\mathcal{T}}\,.$

On the other hand, considering the variable
$\mathcal{N}_{\mathcal{T}}(k)/(L_{\mathcal{T}}(S)-S_{\mathcal{T}})\leq 1$ for
$k\leq 0$ and summing over all bonds, we obtain

(3.32) $\sum_{k\leq
0}\left(\mathcal{N}_{\mathcal{T}}(k)\right)^{{\gamma}}\geq(L_{\mathcal{T}}(S)-S_{\mathcal{T}})^{{\gamma}}\,.$

These inequalities will be used to provide upper and lower bounds for the
generating function of the cost process. Other useful probabilistic results
are postponed to Appendix B.

###### Proof of Theorem 2.2.

As underlined in [7], the introduction of the local times $N_{n}(k)$’s
provides an interpretation of the collision times
$(T_{n})_{n\in{\mathbb{N}}_{0}}$ (defined in (2.12)) as a random walk in
random scenery on bonds. More generally, the cost process of the form (2.6)
satisfies the following identity

(3.33) $\displaystyle C_{\mathcal{T}}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}\left[\mathcal{N}_{\mathcal{T}}^{+}(k){\zeta}_{k}^{+}+\mathcal{N}_{\mathcal{T}}^{-}(k){\zeta}_{k}^{-}\right]$
(3.34) $\displaystyle=\sum_{k\leq
0}\mathcal{N}_{\mathcal{T}}(k){\zeta}_{k}^{0}+\sum_{k>0}\mathcal{N}_{\mathcal{T}}(k){\zeta}_{k}^{+}\,.$

Since the local times are functions of $S$ only, it turns out that, given $S$,
$C_{\mathcal{T}}$ is a sum of independent random variables.

Using (3.34), we can rearrange the terms inside the characteristic function of
$C_{\mathcal{T}}$ and get, for $s\in{\mathbb{R}}$,

$\displaystyle\mathbb{E}\left[e^{isC_{\mathcal{T}}}\right]$
$\displaystyle=\mathbb{E}\left[\exp\left(\>is\sum_{k=1}^{\mathcal{T}}\eta_{k}\>\right)\right]=\mathbb{E}\left[e^{is\left[\sum_{k\leq
0}\mathcal{N}_{\mathcal{T}}(k){\zeta}_{k}^{0}+\sum_{k>0}\mathcal{N}_{\mathcal{T}}(k){\zeta}_{k}^{+}\right]}\right]$
$\displaystyle=\mathbb{E}\left[\left.\mathbb{E}\left[\exp\left(\>is\sum_{k\leq
0}\mathcal{N}_{\mathcal{T}}(k){\zeta}_{k}^{0}\>\right)\exp\left(\>is\sum_{k>0}\mathcal{N}_{\mathcal{T}}(k){\zeta}_{k}^{+}\>\right)\right|S\right]\right]$
(3.35) $\displaystyle=\mathbb{E}\left[\prod_{k\leq
0}\phi_{{\zeta}_{1}^{0}}(s\mathcal{N}_{\mathcal{T}}(k))\cdot\left(\phi_{{\zeta}_{1}^{+}}(s)\right)^{S_{\mathcal{T}}}\right]\,,$

where in the last line we used the conditional independence mentioned above.

##### Case $\hat{\gamma}_{0}=+\infty\>\implies\>{\rho}_{+}<{\rho}_{0}\,$.

Here we get a direct result, since local times disappear. Explicitly, we have

$\phi_{C_{\mathcal{T}}}(s)={\mathbb{E}}\left[\left(\phi_{{\zeta}_{1}^{+}}(s)\right)^{S_{\mathcal{T}}}\right]=\begin{cases}{\mathbb{E}}\left[e^{-s^{\hat{\gamma}_{+}}[\Re(c_{+})+o(1)]]S_{\mathcal{T}}}\cdot
e^{is^{\hat{\gamma}_{+}}[-\Im(c_{+})+o(1)]]S_{\mathcal{T}}}\right]\>&\mbox{if
}\hat{\gamma}_{+}\in(0,1)\,,\\\
{\mathbb{E}}\left[e^{is[{\mu}_{+}+o(1)]S_{\mathcal{T}}}\cdot
e^{-s^{{\gamma}_{+}}[\Re(c_{+})+o(1)]]S_{\mathcal{T}}}\right]\>&\mbox{if
}\hat{\gamma}_{+}=1\,,\end{cases}$

hence by applying Lemma B.1 we immediately obtain

$K_{low}\cdot
x^{-\hat{\gamma}_{+}{\beta}/2}\leq\mathbb{P}[|C_{\mathcal{T}}(Y)|>x]\leq
K_{up}\cdot x^{-\hat{\gamma}_{+}{\beta}/2}\,,$

with $K_{low}=K_{up}$ if $\hat{\gamma}_{+}=1$ or $\Im(c_{+})=0$ .

In the following two paragraphs we will move to the generating function
formalism, which is justified by the additional hypothesis
${\zeta}_{1}^{+},{\zeta}_{1}^{0}\geq 0$. In particular, we will provide upper
and lower bounds on the generating function

$\mathcal{G}_{C_{\mathcal{T}}}(s)\coloneqq{\mathbb{E}}[e^{-sC_{\mathcal{T}}}]\,,$

that will allow us to determine, respectively, lower and upper bounds for the
tails of the random variable $C_{\mathcal{T}}(Y)$.

We anticipate that this analysis requires, in some cases, the stability of
${\zeta}_{1}^{0}$. In the presence of a generic domain of attraction, the
management of a random error can be technically difficult to control.

##### Case $\hat{\gamma}_{0}=1\,$ (equiv. ${\gamma}_{0}\in(1,2]$).

We start from Eq. (3.2), that is

(3.36)
$\mathcal{G}_{C_{\mathcal{T}}}(s)={\mathbb{E}}[e^{-sC_{\mathcal{T}}}]=\mathbb{E}\left[\prod_{k\leq
0}\mathcal{G}_{{\zeta}_{1}^{0}}(s\mathcal{N}_{\mathcal{T}}(k))\cdot\left(\mathcal{G}_{{\zeta}_{1}^{+}}(s)\right)^{S_{\mathcal{T}}}\right]\,,\qquad
s\geq 0\,.$

Observing that $s\mathcal{N}_{\mathcal{T}}(k)\to 0$ pointwise as $s\to 0$, we
can replace the generating functions with their expansions around $0$

(3.37)
$\displaystyle\mathcal{G}_{{\zeta}_{1}^{0}}(\theta)=1-{\mu}_{0}\theta+o(\theta)\,,\qquad$
$\displaystyle\mbox{as}\quad\theta\to 0^{+}\,,$ (3.38)
$\displaystyle\mathcal{G}_{{\zeta}_{1}^{+}}(\theta)=1-\tilde{\mu}_{+}\theta^{\hat{\gamma}_{+}}+o(\theta^{\hat{\gamma}_{+}})\qquad$
$\displaystyle\mbox{as}\quad\theta\to 0^{+}\,,$

where

$\tilde{\mu}_{+}=\begin{cases}\frac{\Re(c_{+})}{\cos(\pi\hat{{\gamma}}_{+}/2)}\qquad&\mbox{if}\quad\hat{\gamma}_{+}\in(0,1)\,,\\\
{\mu}_{+}\qquad&\mbox{if}\quad\hat{\gamma}_{+}=1\,,\end{cases}$

according to (3.28) and (3.27). Hence we obtain

$\mathcal{G}_{C_{\mathcal{T}}}(s)=\mathbb{E}\left[e^{-{\mu}_{0}s(L_{\mathcal{T}}-S_{\mathcal{T}})+\sum_{k\leq
0}o(s\mathcal{N}_{\mathcal{T}}(k))}\cdot
e^{-\tilde{\mu}_{+}(s^{\hat{\gamma}_{+}}+o(s^{\hat{\gamma}_{+}}))S_{\mathcal{T}}}\right]\,,$

and we can apply Lemma B.1 by defining

$Z_{1}\coloneqq e^{-{\mu}_{0}s(L_{\mathcal{T}}-S_{\mathcal{T}})+\sum_{k\leq
0}o(s\mathcal{N}_{\mathcal{T}}(k))}\qquad\mbox{ and }\qquad Z_{2}\coloneqq
e^{-s^{\hat{\gamma}_{+}}(\tilde{\mu}_{+}+o(1))S_{\mathcal{T}}}\,.$

${\mathbb{E}}[Z_{2}]$ is trivially asymptotically equivalent to
$\mathcal{G}_{S_{\mathcal{T}}}(\tilde{\mu}_{+}s^{\hat{\gamma}_{+}})=1-c_{2}s^{{\gamma}_{2}}+o(s^{{\gamma}_{2}})$,
with ${\gamma}_{2}=\hat{\gamma}_{+}{\beta}/2$ and $c_{2}$ given by Corollary
3.10, so we have to focus our efforts on the random variable $Z_{1}$. First of
all, notice that by definition $\forall\epsilon>0\,,\>\exists\bar{s}_{k}>0$
such that $\forall s\leq\bar{s}_{k}$ we have
$|o(s\mathcal{N}_{\mathcal{T}}(k))|\leq\epsilon
s\mathcal{N}_{\mathcal{T}}(k)$. As a consequence, we can write

$\forall s\leq\min\\{(\bar{s}_{k})_{k\leq 0}\\}\,,\qquad\left|\sum_{k\leq
0}o(s\mathcal{N}_{\mathcal{T}}(k))\right|\leq\sum_{k\leq
0}\left|o(s\mathcal{N}_{\mathcal{T}}(k))\right|\leq\epsilon
s(L_{\mathcal{T}}-S_{\mathcal{T}})\,,$

and conclude that $\sum_{k\leq
0}o(s\mathcal{N}_{\mathcal{T}}(k))=o(s(L_{\mathcal{T}}-S_{\mathcal{T}}))$.
Using this observation, we can infer that the expansion with respect to $s$ of
${\mathbb{E}}[Z_{1}]$ has the same leading order of

(3.39)
${\mathbb{E}}[\mathcal{G}_{V}(s(L_{\mathcal{T}}-S_{\mathcal{T}}))]=\mathcal{G}_{V\cdot(L_{\mathcal{T}}-S_{\mathcal{T}})}(s)\,,$

where $V$ is a random variable independent of
$L_{\mathcal{T}}-S_{\mathcal{T}}$ with ${\mathbb{E}}[V]={\mu}_{0}$ and tail
behavior determined by the little-o. The equality in Eq. (3.39) is due to the
fact that if $V,\,W$ are independent random variables, by means of the law of
total expectation we can write

$\mathcal{G}_{V\cdot W}(s)={\mathbb{E}}[e^{-sV\cdot
W}]={\mathbb{E}}[{\mathbb{E}}[e^{-sV\cdot
W}|W]]={\mathbb{E}}[\mathcal{G}_{V}(sW)]={\mathbb{E}}[\mathcal{G}_{W}(sV)]\,.$

Starting from Eq. (3.39), we can therefore affirm that the asymptotic behavior
of $Z_{1}$ is ruled by $L_{\mathcal{T}}-S_{\mathcal{T}}$, that is the random
variable with slower tail decay. Then

${\mathbb{E}}[Z_{1}]=1-c_{1}s^{{\gamma}_{1}}+o(s^{{\gamma}_{1}})\,,$

with ${\gamma}_{1}=\hat{\beta}/2$, except for the case
${\beta}\equiv\hat{\beta}=1$ where the constant $c_{1}$ is replaced by a
slowly varying function $c_{1}\sqrt{\log(1/s)}$. Formally, this conclusion
stems from the application of Lemma B.2.

###### Upper bound:

Using the estimates in Lemma B.1, we can finally write

$\mathcal{G}_{C_{\mathcal{T}}}(s)\leq{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]+\sqrt{\hbox{\rm
Var}(Z_{1})\hbox{\rm
Var}(Z_{2})}=1-c_{up}s^{\min\\{{\rho}_{0},{\rho}_{+}\\}/2}+o(s^{\min\\{{\rho}_{0},{\rho}_{+}\\}/2})\,,$

given that $\sqrt{\hbox{\rm Var}(Z_{1})\hbox{\rm
Var}(Z_{2})}=\sqrt{(2-2^{{\rho}_{0}/2})(2-2^{{\rho}_{+}/2})c_{1}c_{2}}s^{\frac{{\rho}_{0}+{\rho}_{+}}{2}}$,
with an additional logarithmic factor if ${\rho}_{0}\leq{\rho}_{+}$ and
${\beta}\equiv\hat{\beta}=1$.

###### Lower bound:

Similarly, by means of the reversed inequality, we can state that

$\mathcal{G}_{C_{\mathcal{T}}}(s)\geq{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]-\sqrt{\hbox{\rm
Var}(Z_{1})\hbox{\rm
Var}(Z_{2})}=1-c_{low}s^{\min\\{{\rho}_{0},{\rho}_{+}\\}/2}+o(s^{\min\\{{\rho}_{0},{\rho}_{+}\\}/2})\,,$

with the same logarithmic correction as before.

In terms of the tail asymptotic for the cost process, we have therefore
obtained

$K_{low}(x)\cdot x^{-{\rho}/2}\leq{\mathbb{P}}[C_{\mathcal{T}}(Y)>x]\leq
K_{up}(x)\cdot x^{-{\rho}/2}\,,$

with matching slowly varying functions $K_{low}(x)=K_{up}(x)$ when
${\rho}_{+}\neq{\rho}_{0}$ or ${\rho}_{+}={\rho}_{0}$ and
${\beta}\equiv\hat{\beta}=1$. Notice that in most of the cases
$K_{low},K_{up}$ are simply real positive constants: it is enough to avoid the
limiting case ${\beta}\equiv\hat{\beta}=1$.

##### Case $\hat{\gamma}_{0}\in(0,1)$.

We proceed as before, substituting Eq. (3.38) and

$\mathcal{G}_{{\zeta}_{1}^{0}}({\theta})=1-\tilde{c}_{0}{\theta}^{\hat{\gamma}_{0}}+o({\theta}^{\hat{\gamma}_{0}})\,,\qquad\mbox{with}\quad\tilde{c}_{0}\coloneqq\frac{\Re(c_{0})}{\cos(\pi\hat{{\gamma}}_{0}/2)}\,,\quad\hat{{\gamma}}_{0}\in(0,1)\,,\quad\mbox{as}\quad{\theta}\to
0^{+}\,,$

in Eq. (3.36), and we get

$\mathcal{G}_{C_{\mathcal{T}}}(s)=\mathbb{E}\left[e^{-\tilde{c}_{0}s^{\hat{{\gamma}}_{0}}\sum_{k\leq
0}(\mathcal{N}_{\mathcal{T}}(k))^{\hat{{\gamma}}_{0}}+o(s^{\hat{{\gamma}}_{0}}\sum_{k\leq
0}(\mathcal{N}_{\mathcal{T}}(k))^{\hat{{\gamma}}_{0}})}\cdot
e^{-\tilde{\mu}_{+}(s^{\hat{{\gamma}}_{+}}+o(s^{\hat{{\gamma}}_{+}}))S_{\mathcal{T}}}\right]\eqqcolon{\mathbb{E}}[Z_{1}\cdot
Z_{2}]\,,$

where $Z_{2}$ is the same random variable as in the previous paragraph. In
order to provide an asymptotic expansion for ${\mathbb{E}}[Z_{1}]$ and
$\hbox{\rm Var}(Z_{1})$, we can equivalently study
${\mathbb{E}}[\mathcal{G}_{V}(sW)]=\mathcal{G}_{V\cdot W}(s)\,,$ with
$V={\zeta}_{1}^{0}$ and $W=\left(\sum_{k\leq
0}(\mathcal{N}_{\mathcal{T}}(k))^{\hat{{\gamma}}_{0}}\right)^{1/\hat{{\gamma}}_{0}}$.
We have not direct information about the random variable $W$, but using the
estimates in (3.31) and (3.32) on local times, Lemma 3.11 and Appendix B.1, we
get

$w^{-\hat{\beta}/2}\propto{\mathbb{P}}[L_{\mathcal{T}}-S_{\mathcal{T}}>w]\leq{\mathbb{P}}[W>w]\leq{\mathbb{P}}[(L_{\mathcal{T}}-S_{\mathcal{T}})^{1/\hat{{\gamma}}_{0}}>w]\propto
w^{-\hat{{\gamma}}_{0}\hat{\beta}/2}\,,$

with an additional term $\sqrt{\log(w)}$ on both sides when
${\beta}\equiv\hat{\beta}=1$. Consequently, by applying Lemma B.2 to the
products $V\cdot(L_{\mathcal{T}}-S_{\mathcal{T}})$ and
$V\cdot(L_{\mathcal{T}}-S_{\mathcal{T}})^{1/\hat{{\gamma}}_{0}}$, we obtain
the bounds

(3.40a) $\displaystyle{\mathbb{E}}[Z_{1}]$ $\displaystyle\leq
1-c_{1}^{+}s^{\min\\{\hat{{\gamma}}_{0},\hat{{\beta}}/2\\}}+o(s^{\min\\{\hat{{\gamma}}_{0},\hat{{\beta}}/2\\}})\,,$
(3.40b) $\displaystyle{\mathbb{E}}[Z_{1}]$ $\displaystyle\geq
1-c_{1}^{-}s^{\hat{{\gamma}}_{0}\hat{{\beta}}/2}+o(s^{\hat{{\gamma}}_{0}\hat{{\beta}}/2})\,,$

that imply also

$\hbox{\rm Var}(Z_{1})={\mathbb{E}}[Z_{1}^{2}]-{\mathbb{E}}[Z_{1}]^{2}\leq
2c_{1}^{-}s^{\hat{{\gamma}}_{0}\hat{{\beta}}/2}+o(s^{\hat{{\gamma}}_{0}\hat{{\beta}}/2})\,.$

More precisely, if $\hat{{\gamma}}_{0}=\hat{{\beta}}/2$, we have to introduce
a logarithmic correction in the upper bound for the expectation of $Z_{1}$,
which means ${\mathbb{E}}[Z_{1}]\leq
1-c_{1}^{+}s^{\hat{{\gamma}}_{0}}\log(1/s)+o(s^{\hat{{\gamma}}_{0}}\log(1/s))$.
If $\hat{{\gamma}}_{0}=\hat{{\beta}}/2$ and ${\beta}\equiv\hat{\beta}=1$, the
slowly varying function is $c_{1}^{+}\log^{3/2}(1/s)$, whereas if
$\hat{\gamma}_{0}>1/2$ and ${\beta}\equiv\hat{\beta}=1$ we have
$c_{1}^{+}\sqrt{\log(1/s)}$. With regard to the lower bound and the estimate
on the variance, the square root of the logarithm appears whenever
${\beta}\equiv\hat{\beta}=1$.

###### Lower bound:

By means of Lemma B.1, we find that

$\displaystyle\mathcal{G}_{C_{\mathcal{T}}}(s)$
$\displaystyle\geq{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]-\sqrt{\hbox{\rm
Var}(Z_{1})\hbox{\rm Var}(Z_{2})}$ $\displaystyle\geq
1-c_{1}^{-}s^{\hat{{\gamma}}_{0}\hat{{\beta}}/2}-c_{2}s^{\hat{{\gamma}}_{+}{\beta}/2}-\sqrt{2(2-2^{\hat{{\gamma}}_{+}{\beta}/2})c_{1}^{-}c_{2}}s^{\frac{\hat{{\gamma}}_{0}\hat{{\beta}}+\hat{{\gamma}}_{+}{\beta}}{4}}+\dots\,,$

with $c_{1}^{-}$ replaced by $c_{1}^{-}\sqrt{\log(1/s)}$ if
${\beta}=\hat{\beta}=1$, which results in an upper bound on the tail of the
cost process

${\mathbb{P}}[C_{\mathcal{T}}(Y)>x]\leq K_{up}(x)\cdot x^{-{\rho}/2}\,,$

with $K_{up}(x)$ slowly varying function at infinity. Note that
$K_{up}(x)=K_{up}\sqrt{\log(x)}$ if ${\beta}=\hat{\beta}=1$ and
${\rho}_{0}\leq{\rho}_{+}$, whereas it is simply a real positive constant in
all the other cases.

###### Upper bound:

This time the estimates (3.40), combined with the Cauchy-Schwarz inequality,
are not always adequate to provide a meaningful bound. In fact, bearing in
mind the previous clarification concerning the limiting cases, for which one
needs to make some small changes in terms of slowly varying functions, we have

$\displaystyle\mathcal{G}_{C_{\mathcal{T}}}(s)$
$\displaystyle\leq{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]+\sqrt{\hbox{\rm
Var}(Z_{1})\hbox{\rm Var}(Z_{2})}$
$\displaystyle\leq\min\left\\{1,1-c_{1}^{+}s^{\min\\{\hat{{\gamma}}_{0},\hat{{\beta}}/2\\}}-c_{2}s^{\hat{{\gamma}}_{+}{\beta}/2}+\sqrt{2(2-2^{\hat{{\gamma}}_{+}{\beta}/2})c_{1}^{-}c_{2}}s^{\frac{\hat{{\gamma}}_{0}\hat{{\beta}}+\hat{{\gamma}}_{+}{\beta}}{4}}+\dots\right\\}\,,$

which turns into

${\mathbb{P}}[C_{\mathcal{T}}(Y)>x]\geq\begin{cases}K_{low}\cdot
x^{-{\rho}/2}\qquad&\mbox{if}\quad{\rho}\equiv{\rho}_{+}<{\rho}_{0}\,,\\\
K_{low}\cdot
x^{-\min\\{\hat{{\gamma}}_{0},\hat{{\beta}}/2\\}}\qquad&\mbox{if}\quad{\rho}\equiv{\rho}_{0}<{\rho}_{+}\,,\quad\min\\{\hat{{\gamma}}_{0},\hat{{\beta}}/2\\}<({\rho}_{0}+{\rho}_{+})/4\,,\\\
K_{low}\cdot
x^{-{\rho}/2}\qquad&\mbox{if}\quad{\rho}\equiv{\rho}_{+}={\rho}_{0}\,,\quad
c_{2}-\sqrt{2c_{1}^{-}c_{2}}>0\,,\\\ 0\qquad&\mbox{otherwise}\,.\end{cases}$

In particular, notice that this analysis substantiates Remark 3.14.

In order to guarantee the global existence of a substantial lower bound, we
have to impose the stability condition on the law of ${\zeta}_{k}^{0}$’s. Let
us assume that
$\mathcal{G}_{{\zeta}_{1}^{0}}({\theta})=e^{-\tilde{c}_{0}{\theta}^{\hat{{\gamma}}_{0}}-{\mu}_{0}{\theta}}$,
with ${\mu}_{0}\geq 0$. Then the proof for the upper bound simply becomes

$\mathcal{G}_{C_{\mathcal{T}}}(s)\leq{\mathbb{E}}[e^{-\tilde{c}_{0}s^{\hat{{\gamma}}_{0}}(L_{\mathcal{T}}-S_{\mathcal{T}})^{\hat{{\gamma}}_{0}}}\cdot
e^{-s^{\hat{{\gamma}}_{+}}(\tilde{\mu}_{f_{+}}+o(1))S_{\mathcal{T}}}]\>\implies\>{\mathbb{P}}[C_{\mathcal{T}}(Y)>x]\geq
K_{low}(x)\cdot
x^{-\min\left\\{\hat{{\gamma}}_{0},\frac{\hat{{\beta}}}{2},\frac{{\rho}_{+}}{2}\right\\}}\,,$

making the tail asymptotic always non-trivial. More precisely,
$K_{low}(x)=K_{low}\sqrt{\log(x)}$ when the exponent is $\hat{\beta}/2$ with
${\beta}=\hat{\beta}=1$, and $K_{low}(x)=K_{low}\in\mathbbm{R}^{+}$ otherwise
(it is explicit only for the tail exponent ${\rho}_{+}/2$).
On the other hand, the lower bound merely verifies the above result of more
general validity concerning generic domains of attraction:

$\displaystyle\mathcal{G}_{C_{\mathcal{T}}}(s)$
$\displaystyle\geq{\mathbb{E}}[e^{-\tilde{c}_{0}s^{\hat{{\gamma}}_{0}}(L_{\mathcal{T}}-S_{\mathcal{T}})-{\mu}_{0}s(L_{\mathcal{T}}-S_{\mathcal{T}})}\cdot
e^{-s^{\hat{{\gamma}}_{+}}(\tilde{\mu}_{+}+o(1))S_{\mathcal{T}}}]$
$\displaystyle={\mathbb{E}}[e^{-s^{\hat{{\gamma}}_{0}}(\tilde{c}_{0}+o(1))(L_{\mathcal{T}}-S_{\mathcal{T}})}\cdot
e^{-s^{\hat{{\gamma}}_{+}}(\tilde{\mu}_{+}+o(1))S_{\mathcal{T}}}]\,,$
$\displaystyle{\mathbb{P}}[C_{\mathcal{T}}(Y)>x]$ $\displaystyle\leq
K_{up}(x)\cdot x^{-{\rho}/2}\,.$

As a final point, we generalize all these asymptotic results to the law of
ladder costs. Let $(\mathcal{T}_{n})_{n\geq 0}$ be the sequence of ladder
times of the random walk $S$ defined in the introduction. Notice that, for all
$n\geq 1$, the equality (3.33) still holds by replacing $\mathcal{T}$ with
$\mathcal{T}_{n}$, together with the identity
$L_{\mathcal{T}_{n}}(S)=\sum_{k\in{\mathbb{Z}}}\mathcal{N}_{\mathcal{T}_{n}}(k)$.
Moreover,
$S_{\mathcal{T}_{n}}=\sum_{k>0}\mathbbm{1}_{(0\,,\,S_{\mathcal{T}_{n}}]}(k)\,$.
In order to recast the characteristic or generating function of the cost
$C_{\mathcal{T}_{n}}$ in a convenient manner, we need to introduce a further
definition concerning the local times. Let $\mathcal{N}_{(t_{0},t_{f}]}(k)$ be
the number of crossings of $[k-1,k]$ observed in a specified time window
$(t_{0},t_{f}]$, that is

(3.41)
$\mathcal{N}_{(t_{0},t_{f}]}(k)\coloneqq\\#\\{j\in\\{t_{0}+1,\ldots,t_{f}\\}\,:\,[k-1,k]\subseteq[S_{j-1},S_{j}]\\}\,.$

Hence we get

$\displaystyle C_{\mathcal{T}_{n}}$
$\displaystyle=\sum_{k\in\mathbbm{Z}}\left[\mathcal{N}_{\mathcal{T}_{n}}^{+}(k){\zeta}_{k}^{+}+\mathcal{N}_{\mathcal{T}_{n}}^{-}(k){\zeta}_{k}^{-}\right]$
$\displaystyle=\sum_{k\leq
0}\mathcal{N}_{(0,\mathcal{T}_{1}]}(k){\zeta}_{k}^{0}+\sum_{k\leq
S_{\mathcal{T}_{1}}}\mathcal{N}_{(\mathcal{T}_{1},\mathcal{T}_{2}]}(k){\zeta}_{k}^{0}+\dots+\sum_{k\leq
S_{\mathcal{T}_{n-1}}}\mathcal{N}_{(\mathcal{T}_{n-1},\mathcal{T}_{n}]}(k){\zeta}_{k}^{0}$
$\displaystyle\hskip
42.67912pt+\sum_{k>0}\mathcal{N}_{(0,\mathcal{T}_{1}]}(k){\zeta}_{k}^{+}+\sum_{k>S_{\mathcal{T}_{1}}}\mathcal{N}_{(\mathcal{T}_{1},\mathcal{T}_{2}]}(k){\zeta}_{k}^{+}+\dots+\sum_{k>S_{\mathcal{T}_{n-1}}}\mathcal{N}_{(\mathcal{T}_{n-1},\mathcal{T}_{n}]}(k){\zeta}_{k}^{+}$
$\displaystyle=\sum_{k\leq
0}\mathcal{N}_{\mathcal{T}_{n}}(k){\zeta}_{k}^{0}+\sum_{k\in(0,S_{\mathcal{T}_{1}}]}\mathcal{N}_{(\mathcal{T}_{1},\mathcal{T}_{n}]}(k){\zeta}_{k}^{0}+\dots+\sum_{k\in(S_{\mathcal{T}_{n-2}},S_{\mathcal{T}_{n-1}}]}\mathcal{N}_{(\mathcal{T}_{n-1},\mathcal{T}_{n}]}(k){\zeta}_{k}^{0}$
$\displaystyle\hskip
42.67912pt+\sum_{k>0}\mathbbm{1}_{(0,S_{\mathcal{T}_{n}}]}(k){\zeta}_{k}^{+}\,,$

and also

$\displaystyle\mathcal{G}_{C_{\mathcal{T}_{n}}}(s)$
$\displaystyle\coloneqq{\mathbb{E}}\left[e^{-sC_{\mathcal{T}_{n}}}\right]=\mathbb{E}\left[\left.{\mathbb{E}}\left[e^{-s\sum_{k\in\mathbbm{Z}}\left[\mathcal{N}_{\mathcal{T}_{n}}^{+}(k){\zeta}_{k}^{+}+\mathcal{N}_{\mathcal{T}_{n}}^{-}(k){\zeta}_{k}^{-}\right]}\right|S\right]\right]$
$\displaystyle=\mathbb{E}\left[\prod_{k\leq
0}\mathcal{G}_{{\zeta}^{0}_{1}}(s\mathcal{N}_{\mathcal{T}_{n}}(k))\prod_{k\in(0,S_{\mathcal{T}_{1}}]}\mathcal{G}_{{\zeta}^{0}_{1}}(s\mathcal{N}_{(\mathcal{T}_{1},\mathcal{T}_{n}]}(k))\dots\right.$
$\displaystyle\hskip
85.35826pt\left.\dots\prod_{k\in(S_{\mathcal{T}_{n-2}},S_{\mathcal{T}_{n-1}}]}\mathcal{G}_{{\zeta}^{0}_{1}}(s\mathcal{N}_{(\mathcal{T}_{n-1},\mathcal{T}_{n}]}(k))\cdot\left(\mathcal{G}_{{\zeta}^{+}_{1}}(s)\right)^{S_{\mathcal{T}_{n}}}\right]\,.$

Then, when we consider upper and lower bounds for
$\mathcal{G}_{C_{\mathcal{T}_{n}}}(s)$, the relevant quantities to deal with
are $S_{\mathcal{T}_{n}}$ and

$\sum_{k\leq
0}\mathcal{N}_{\mathcal{T}_{n}}(k)+\sum_{k\in(0,S_{\mathcal{T}_{1}}]}\mathcal{N}_{(\mathcal{T}_{1},\mathcal{T}_{n}]}(k)+\dots+\sum_{k\in(S_{\mathcal{T}_{n-2}},S_{\mathcal{T}_{n-1}}]}\mathcal{N}_{(\mathcal{T}_{n-1},\mathcal{T}_{n}]}(k)=L_{\mathcal{T}_{n}}-S_{\mathcal{T}_{n}}\,.$

More precisely, we are interested in their generating functions. Due to the
renewal structure of the processes $(S_{\mathcal{T}_{n}})_{n\geq 0}$ and
$(L_{\mathcal{T}_{n}}(S))_{n\geq 0}$, the ladder random variables can be seen
as the sum of $n$ i.i.d. first-ladder quantities. Thus, the previous results
can be immediately generalized: we just have to introduce a multiplicative
factor $n$, which stems from the factorization of the expectations, in front
of the slowly varying functions $K(x)$, $K_{up}(x)$ and $K_{low}(x)\,$. ∎

###  3.3. Results for the random walks in random media Y

Consider the random walk on random medium $Y$ defined in Eq. (2.9), under the
hypothesis that the underlying random walk $S$ has symmetric i.i.d. increments
$(\xi_{k})_{k\in{\mathbb{N}}}$. As already mentioned, the first-ladder height
$Y_{\mathcal{T}}$ and the length $L_{\mathcal{T}}(Y)$ can be equivalently
interpreted as first-ladder costs expressed as RWRSB with appropriate scenery.
We then use our main Theorem 3.13, together with some simplifications
occurring in this setting, to prove the Corollaries 2.3, 2.4 and 2.5.

In the following, we will refer to the notation introduced in previous
sections, except for Eq.(3.27) that can be slightly simplified by considering
only

$\displaystyle\phi_{{\zeta}_{1}}({\theta})$
$\displaystyle=\begin{cases}1-ce^{-i\frac{\pi}{2}{\gamma}}{\theta}^{{\gamma}}+o({\theta}^{{\gamma}})\,,\quad&{\gamma}=\hat{{\gamma}}\in(0,1)\,;\>c\in\mathbbm{R}^{+}\,,\\\
1+i{\mu}{\theta}+o({\theta})\,,\quad&{{\gamma}}\in(1,2]\,,\hat{{\gamma}}=1\,;\>{\mu}\in\mathbbm{R}^{+}\,.\end{cases}$

####  3.3.1.  First-ladder height $Y_{\mathcal{T}}$.

###### Proof of Corollary 2.3.

Recall that $Y_{\mathcal{T}}$ can be seen as the value at time $\mathcal{T}$
of a RWRSB driven by $S$ and with scenery ${\zeta}^{+}=-{\zeta}^{-}={\zeta}$.
Notice that with this choice
$\hat{{\gamma}}\equiv\hat{{\gamma}}_{+}=\hat{{\gamma}}_{-}$, and
$\hat{{\gamma}}_{0}=+\infty\,$ implying that ${\rho}_{+}<{\rho}_{0}\,$.
Therefore Theorem 3.13 applies.

Moreover, since ${\zeta}$ are positive random variables (and thus also
$Y_{\mathcal{T}}$ is positive), we can replace the characteristic functions
with the generating functions in the appropriate part of the proof of the main
theorem. This enable us to find explicit constants and get:

  * •

If ${\gamma}\in(1,2]$, which means $\hat{{\gamma}}=1$, then
${\mathbb{E}}[e^{-sY_{\mathcal{T}}}]={\mathbb{E}}[e^{-s({\mu}+o(1))S_{\mathcal{T}}}]$
and $Y_{\mathcal{T}}$ is in the basin of attraction of a stable law with
parameter ${\beta}/2\,$. Explicitly, by applying Corollary 3.10, we get

(3.42)
${\mathbb{P}}(Y_{\mathcal{T}}>x)\sim\frac{\sqrt{\nu}\mu^{{\beta}/2}}{{\Gamma}(1-{\beta}/2)}\,{\Phi}(1,0)\,x^{-{\beta}/2}\,,\qquad\mbox{as
}x\to\infty\,\,.$

As previously mentioned, the limiting case ${\beta}=2$ is ensured by [12,
Theorem 4].

  * •

If $\gamma\in(0,1)$, then $\hat{{\gamma}}=\gamma$ and we have
${\mathbb{E}}[e^{-sY_{\mathcal{T}}}]={\mathbb{E}}[e^{-s^{\gamma}(c+o(1))S_{\mathcal{T}}}]$
from which

(3.43)
${\mathbb{P}}(Y_{\mathcal{T}}>x)\sim\frac{\sqrt{\nu}c^{{\beta}/2}}{{\Gamma}(1-{\gamma}{\beta}/2)}\,{\Phi}(1,0)\,x^{-{\gamma}{\beta}/2}\,,\qquad\mbox{as
}x\to\infty\,\,.$

∎

####  3.3.2.  First-ladder length.

###### Proof of Corollary 2.4.

Recall that $L_{\mathcal{T}}(Y)$ can be seen as the value at time
$\mathcal{T}$ of a RWRSB driven by $S$ and with scenery
${\zeta}^{+}={\zeta}^{-}={\zeta}$. Notice that with this choice
$\hat{\gamma}\equiv\hat{{\gamma}}_{+}=\hat{{\gamma}}_{-}=\hat{{\gamma}}_{0}$
and therefore ${\rho}_{+}\geq{\rho}_{0}\,$. Following Theorem 3.13, we derive
the next distinct cases.

  * •

If ${\gamma}\in(1,2]\,$, that is $\hat{\gamma}_{0}=\hat{\gamma}_{+}=1$, we
obtain:

    1. (i)

If $E(|\xi_{1}|)<\infty$, which means ${\beta}\in(1,2]$ and
${\rho}_{0}<{\rho}_{+}$, then $L_{\mathcal{T}}(Y)$ is in the basin of
attraction of a stable law with parameter $1/2$,

${\mathbb{P}}(L_{\mathcal{T}}(Y)>x)\sim K\cdot x^{-1/2}\,,\qquad\mbox{as
}x\to\infty\,\,;$

    2. (ii)

If $\hat{\beta}\equiv{\beta}=1$, that is ${\rho}_{0}={\rho}_{+}=1$, then

${\mathbb{P}}(L_{\mathcal{T}}(Y)>x)\sim
K\cdot\sqrt{\log(x)}x^{-1/2}\,,\qquad\mbox{as }x\to\infty\,\,;$

    3. (iii)

If $\hat{\beta}\equiv{\beta}\in(0,1)$, namely
${\rho}_{0}={\rho}_{+}=\hat{\beta}$, then $L_{\mathcal{T}}(Y)$ is in the basin
of attraction of a stable law with parameter $\hat{\beta}/2$,

$K_{low}\cdot x^{-\hat{\beta}/2}\leq{\mathbb{P}}(L_{\mathcal{T}}(Y)>x)\leq
K_{up}\cdot x^{-\hat{\beta}/2}\,,\qquad\mbox{as }x\to\infty\,\,.$

  * •

If ${\gamma}\equiv{\gamma}_{0}={\gamma}_{+}\in(0,1)$, instead, we have to
introduce the additional assumption of stability for the random medium. If we
assume that
$\mathcal{G}_{{\zeta}_{1}}({\theta})=e^{-\tilde{c}{\theta}^{\gamma}-{\mu}{\theta}}$,
with ${\mu}\geq 0$, we can write

${\mathbb{E}}[e^{-\tilde{c}s^{\gamma}(1+\frac{{\mu}}{\tilde{c}}s^{1-{\gamma}})L_{\mathcal{T}}}]\leq\mathcal{G}_{L_{\mathcal{T}}(Y)}(s)\leq{\mathbb{E}}[e^{-\tilde{c}s^{\gamma}(L_{\mathcal{T}}-S_{\mathcal{T}})^{\gamma}}\cdot
e^{-\tilde{c}s^{\gamma}(1+\frac{{\mu}}{\tilde{c}}s^{1-{\gamma}})S_{\mathcal{T}}}]\,.$

In conclusion, in terms of the tail asymptotic for the first-ladder length, we
have

    1. (i)

If $E(|\xi_{1}|)<\infty$, that is ${\beta}\in(1,2]$,

$K_{low}\cdot
x^{-\min\left\\{\frac{1}{2},\frac{{\gamma}{\beta}}{2}\right\\}}\leq{\mathbb{P}}(L_{\mathcal{T}}(Y)>x)\leq\frac{\sqrt{E(|\xi_{1}|)\tilde{c}}}{{\Gamma}(1-{\gamma}/2)}\,{\Phi}(1,0)\,x^{-{\gamma}/2}\,,\qquad\mbox{as
}x\to\infty\,\,;$

    2. (ii)

If $\hat{\beta}\equiv{\beta}=1$, which means ${\rho}_{0}={\rho}_{+}={\gamma}$,
then

$\frac{\sqrt{\nu\tilde{c}}}{{\Gamma}(1-{\gamma}/2)}\,{\Phi}(1,0)\,x^{-\frac{{\gamma}}{2}}\leq{\mathbb{P}}(L_{\mathcal{T}}(Y)>x)\leq\sqrt{\frac{2{\nu}\tilde{c}{\gamma}}{\pi}}\,\frac{{\Phi}(1,0)\,\sqrt{\log(x)}}{\Gamma(1-{\gamma}/2)}\,x^{-{\gamma}/2}\,,\qquad\mbox{as
}x\to\infty\,\,;$

    3. (iii)

If $\hat{\beta}\equiv\beta\in(0,1)$, with ${\rho}_{0}={\rho}_{+}$, as
$x\to\infty$

$\frac{\sqrt{\nu}\tilde{c}^{\hat{\beta}/2}}{{\Gamma}(1-{\gamma}\hat{\beta}/2)}\,{\Phi}(1,0)\,x^{-{\gamma}\hat{\beta}/2}\leq{\mathbb{P}}(L_{\mathcal{T}}(Y)>x)\leq\sqrt{\frac{\nu}{\cos(\pi\hat{\beta}/2)}}\frac{\tilde{c}^{\hat{\beta}/2}}{{\Gamma}(1-{\gamma}\hat{\beta}/2)}\,{\Phi}(1,0)\,x^{-{\gamma}\hat{\beta}/2}\,\,.$

Observe that with this particular choice of RWRSB, we can obtain some more
explicit constants due to the fact that we have dependence on the random
variable $L_{\mathcal{T}}$ instead of the linear combination
$L_{\mathcal{T}}-S_{\mathcal{T}}$ and as a consequence we can apply Corollary
3.7, in addition to Corollary 3.10.

∎

####  3.3.3. Continuous first-passage time for the generalized Lévy-Lorentz
gas

###### Proof of Corollary 2.5.

Recall that the continuous first-passage time
$\mathcal{T}(X)=L_{\mathcal{T}}(Y)-Y_{\mathcal{T}}$ can be seen as the value
at time $\mathcal{T}$ of a RWRSB driven by $S$ and with scenery
${\zeta}^{+}\equiv 0$ and ${\zeta}^{-}=2{\zeta}$. Notice that with this choice
$\hat{{\gamma}}\equiv\hat{{\gamma}}_{-}=\hat{{\gamma}}_{0}<\hat{{\gamma}}_{+}=+\infty$,
and hence
${\rho}=\min\\{{\rho}_{0},{\rho}_{+}\\}={\rho}_{0}=\hat{\gamma}\hat{\beta}$.
As a consequence, Eq. (3.36) simply becomes

$\mathcal{G}_{L_{\mathcal{T}}(Y)-Y_{\mathcal{T}}}(s)=\mathbb{E}\left[\prod_{k\leq
0}\mathcal{G}_{{\zeta}_{1}}(s\mathcal{N}_{\mathcal{T}}(k))\right]\,.$

As before, following Theorem 3.13, we have to study different cases:

  * •

If ${\gamma}\in(1,2]\,$, that is $\hat{{\gamma}}=1$, we obtain the following
results.

    1. (i)

If $E(|\xi_{1}|)<\infty$, with ${\beta}\in(1,2]$ and $\hat{\beta}=1$, then
$\mathcal{T}(X)$ is in the basin of attraction of a $1/2$-stable law,

${\mathbb{P}}(\mathcal{T}(X)>t)\sim K\cdot t^{-1/2}\,,\qquad\mbox{as
}t\to\infty\,\,;$

    2. (ii)

If $\hat{\beta}\equiv{\beta}=1$, then

${\mathbb{P}}(\mathcal{T}(X)>t)\sim
K\cdot\sqrt{\log(t)}t^{-1/2}\,,\qquad\mbox{as }t\to\infty\,\,;$

    3. (iii)

If $\hat{\beta}\equiv\beta\in(0,1)$, similarly $\mathcal{T}(X)$ is in the
basin of attraction of a stable law with parameter $\hat{\beta}/2$,

${\mathbb{P}}(\mathcal{T}(X)>t)\sim K\cdot t^{-\hat{\beta}/2}\,,\qquad\mbox{as
}t\to\infty\,\,.$

  * •

If ${\gamma}=\hat{\gamma}\in(0,1)$, we get

    1. (i)

If $E(|\xi_{1}|)<\infty$, that is ${\beta}\in(1,2]$, we can conclude that

$K_{low}\cdot
t^{-\min\left\\{{\gamma},\frac{1}{2}\right\\}}\leq{\mathbb{P}}(\mathcal{T}(X)>t)\leq
K_{up}(t)\cdot t^{-{\gamma}/2}\,,\qquad\mbox{as }t\to\infty\,\,,$

with $K_{up}(t)=K_{up}\log(t)$ if ${\gamma}=1/2\,$, constant otherwise.

    2. (ii)

If $\hat{\beta}\equiv{\beta}=1$, we obtain

$K_{low}\cdot\sqrt{\log(t)}t^{-\min\left\\{{\gamma},\frac{1}{2}\right\\}}\leq{\mathbb{P}}(\mathcal{T}(X)>t)\leq
K_{up}(t)\cdot t^{-{\gamma}/2}\,,\qquad\mbox{as }t\to\infty\,\,,$

with

$K_{up}(t)=K_{up}\begin{cases}\sqrt{\log(t)}\qquad&\mbox{if}\quad{\gamma}>1/2\,,\\\
\log^{3/2}(t)\qquad&\mbox{if}\quad{\gamma}=1/2\,,\\\
1\qquad&\mbox{if}\quad{\gamma}<1/2\,,\end{cases}$

    3. (iii)

If $\hat{\beta}\equiv\beta\in(0,1)$, we get

$K_{low}\cdot
t^{-\min\left\\{{\gamma},\frac{\hat{\beta}}{2}\right\\}}\leq{\mathbb{P}}(\mathcal{T}(X)>x)\leq
K_{up}\cdot t^{-{\gamma}\hat{\beta}/2}\,,\qquad\mbox{as }t\to\infty\,\,.$

Notice that this specific choice of scenery does not require to restrict the
hypothesis about the random medium to stable random variables, in contrast to
the general statement of Theorem 3.13.

∎

##  Appendix A Lemma

######  Lemma A.1.

Let

(A.1) $\displaystyle P(z,t)=\sum_{n=0}^{\infty}z^{n}p_{n}(t),\qquad$
$\displaystyle Q(z,t)=\sum_{n=0}^{\infty}z^{n}q_{n}(t)\,,$ (A.2)
$\displaystyle P^{*}(z,t)=\sum_{n=0}^{\infty}z^{n}p_{n}^{*}(t),\qquad$
$\displaystyle Q^{*}(z,t)=\sum_{n=0}^{\infty}z^{n}q_{n}^{*}(t)\,,$

where $p_{0}(t)\equiv q_{0}(t)\equiv p_{0}^{*}(t)\equiv q_{0}^{*}(t)\equiv 1$;
and for $n\geq 1$, $p_{n}$ and $p_{n}^{*}$ as functions of $t$ are Fourier
transforms of measures with support in $(0,\infty)$; $q_{n}$ and $q_{n}^{*}$
as functions of $t$ are Fourier transforms of measures in $(-\infty,0].$
Suppose that for some $z_{0}>0$ the four power series converge for $z$ in
$\left(0,z_{0}\right)$ and all real $t,$ and the identity

$P(z,t)Q^{*}(z,t)\equiv P^{*}(z,t)Q(z,t)$

holds there. Then

$P\equiv P^{*},\quad Q\equiv Q^{*}\,.$

###### Proof.

see Section 8.4.1 of [10]. ∎

##  Appendix B Probabilistic tools

###  B.1. Characteristic/generating function of a random variable raised to a
power

Let $X$ be a non-negative random variable and let us consider $Y\coloneqq
X^{\gamma}$, ${\gamma}>0$. We want to determine the characteristic function of
$Y$, keeping in mind that the same reasoning also applies to the generating
function. By means of Tauberian theorems, we know that

(B.1)
$\phi_{X}(s)={\mathbb{E}}\left[e^{isX}\right]=1-ce^{-i\frac{\pi}{2}{\beta}}s^{\beta}+o(s^{\beta})\quad\implies\quad{\mathbb{P}}(X>x)\sim\frac{c}{{\Gamma}(1-{\beta})}x^{-{\beta}}\,.$

Therefore, we immediately obtain

(B.2)
${\mathbb{P}}(Y>y)={\mathbb{P}}(X>y^{1/{\gamma}})\sim\frac{c}{{\Gamma}(1-{\beta})}y^{-{\beta}/{\gamma}}\,.$

By observing that $Y$ has a finite mean if ${\gamma}<{\beta}$, then we can
conclude that

(B.3) $\phi_{Y}(s)=\begin{cases}1+i{\mathbb{E}}[Y]s+o(s)\,\quad&\mbox{if
}{\gamma}<{\beta}\,,\\\
1-c\bar{{\Gamma}}e^{-i\frac{\pi{\beta}}{2{\gamma}}}s^{{\beta}/{\gamma}}+o(s^{{\beta}/{\gamma}})\,\quad&\mbox{if
}{\gamma}\geq{\beta}\,,\end{cases}$

where
$\bar{{\Gamma}}\coloneqq{\Gamma}\left(1-{\beta}/{\gamma}\right)/{\Gamma}\left(1-{\beta}\right)$.

###  B.2. Estimates on joint characteristic/generating functions

######  Lemma B.1.

Assume that, for $k\in\\{1,2\\}$, $Z_{k}(s)$ is a complex random variable
defined by $Z_{k}(s)\coloneqq e^{isX_{k}}$ or $Z_{k}(s)\coloneqq e^{-sX_{k}}$,
whose average therefore corresponds to the characteristic or generating
function of a real or non-negative random variable $X_{k}$.

If $|{\mathbb{E}}[Z_{k}(s)]|=1-c_{k}s^{{\gamma}_{k}}+o(s^{{\gamma}_{k}})$,
with ${\gamma}_{k}\in(0,1)$, $c_{k}\in\mathbbm{R}^{+}$ and $s\to 0^{+}$, then
by defining ${\gamma}\coloneqq\min\\{{\gamma}_{1},{\gamma}_{2}\\}$ and
assuming $c_{1}\neq c_{2}$ if ${\gamma}_{1}={\gamma}_{2}$ and
$\Im(Z_{1}(s)),\Im(Z_{2}(s))>0\,$, we get

$1-k_{+}s^{\gamma}+o(s^{\gamma})\leq|{\mathbb{E}}[Z_{1}Z_{2}(s)]|\leq
1-k_{-}s^{\gamma}+o(s^{\gamma})\,,$

where the positive constants $k_{+}\geq k_{-}$, matching if
${\gamma}_{1}\neq{\gamma}_{2}$, are functions of $c_{1},\,c_{2}$.

###### Proof.

To easy the notation, from now on we will drop the dependence on $s$ of
$Z_{k}$’s. By definition

${\mathbb{E}}[Z_{1}Z_{2}]={\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]+\hbox{\rm
Cov}(Z_{1},\bar{Z}_{2})\,,$

where $\bar{Z}_{2}$ denotes the complex conjugate of $Z_{2}$ and

$\hbox{\rm
Cov}(Z_{1},\bar{Z}_{2})\coloneqq{\mathbb{E}}[(Z_{1}-{\mathbb{E}}[Z_{1}])(Z_{2}-{\mathbb{E}}[Z_{2}])]\,.$

From the Cauchy-Schwarz inequality, we have

$|\hbox{\rm Cov}(Z_{1},\bar{Z}_{2})|\leq\sqrt{\hbox{\rm Var}(Z_{1})\hbox{\rm
Var}(Z_{2})}\,,$

where

$\hbox{\rm
Var}(Z_{k})\coloneqq{\mathbb{E}}[|Z-{\mathbb{E}}[Z]|^{2}]={\mathbb{E}}[|Z_{k}|^{2}]-|{\mathbb{E}}[Z_{k}]|^{2}\,.$

In particular, it holds that

$\displaystyle|{\mathbb{E}}[Z_{1}Z_{2}]|$
$\displaystyle\leq|{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]|+|\hbox{\rm
Cov}(Z_{1},\bar{Z}_{2})|\leq|{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]|+\sqrt{\hbox{\rm
Var}(Z_{1})\hbox{\rm Var}(Z_{2})}\,,$
$\displaystyle|{\mathbb{E}}[Z_{1}Z_{2}]|$
$\displaystyle\geq|{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]|-|\hbox{\rm
Cov}(Z_{1},\bar{Z}_{2})|\geq|{\mathbb{E}}[Z_{1}]{\mathbb{E}}[Z_{2}]|-\sqrt{\hbox{\rm
Var}(Z_{1})\hbox{\rm Var}(Z_{2})}\,.$

Since by assumptions

$\hbox{\rm
Var}(Z_{k})={\mathbb{E}}[|Z_{k}|^{2}]-[1-c_{k}s^{{\gamma}_{k}}+o(s^{{\gamma}_{k}})]^{2}\,,$

to determine the behavior of the variance, as $s\to 0^{+}$, we have to
consider two possible cases:

  * •

If $Z_{k}(s)=e^{-sX_{k}}\,$, then

${\mathbb{E}}[|Z_{k}|^{2}]={\mathbb{E}}[Z_{k}^{2}]={\mathbb{E}}[e^{-2s\,X_{k}}]=1-c_{k}(2s)^{{\gamma}_{k}}+o(s^{{\gamma}_{k}})\,,$

and hence $\hbox{\rm
Var}(Z_{k})=(2-2^{{\gamma}_{k}})c_{k}s^{{\gamma}_{k}}+o(s^{{\gamma}_{k}})\,$,
with $2-2^{{\gamma}_{k}}\in(0,1)\,$;

  * •

If $Z_{k}(s)=e^{isX_{k}}\,$, then ${\mathbb{E}}[|Z_{k}|^{2}]=1$, which implies
$\hbox{\rm Var}(Z_{k})=2c_{k}s^{{\gamma}_{k}}+o(s^{{\gamma}_{k}})\,$.

Let us stress that even if ${\mathbb{E}}[Z_{1}],\,{\mathbb{E}}[Z_{2}]$ are
both characteristic functions, we get

$k_{-}=c_{1}+c_{2}-2\sqrt{c_{1}c_{2}}\geq 0\,,$

with $k_{-}>0$ whenever $c_{1}\neq c_{2}\,$. The statement is therefore
proved. ∎

Notice that Lemma B.1 can be easily generalized to a limiting case that is
useful for the proof of Lemma 3.11 when ${\beta}=2\,$. Indeed, assuming that
${\gamma}_{1}<1\leq{\gamma}_{2}$ and ${\mathbb{E}}[Z_{2}]$ is the
characteristic function of a non-negative random variable $X_{2}$, then

$|{\mathbb{E}}[Z_{1}Z_{2}]|\sim|{\mathbb{E}}[Z_{1}]|=1-c_{1}s^{{\gamma}_{1}}+o(s^{{\gamma}_{1}})\,$

still holds true.

Moreover, if ${\mathbb{E}}[X_{k}]=0$ and we focus on the characteristic
functions, we can extend the result to the range ${\gamma}_{k}\in(0,2]$.

###  B.3. Tail asymptotic of the product of independent random variables

######  Lemma B.2.

Let $V,\,W$ be non-negative independent random variables characterized by the
asymptotic tails

${\mathbb{P}}[V>v]\sim
c_{V}\cdot[\log(v)]^{k_{V}}v^{-{\gamma}_{V}}\,,\qquad{\mathbb{P}}[W>w]\sim
c_{W}\cdot w^{-{\gamma}_{W}}\,,\qquad\mbox{as}\quad v,w\to+\infty\,,$

with ${\gamma}_{V},{\gamma}_{W}\in(0,2)$, $k_{V}\geq 0$ and $c_{V},c_{W}>0$.
It holds that [18]

  * (A)

If ${\gamma}_{V}<{\gamma}_{W}$, then

${\mathbb{P}}[V\cdot W>z]\sim
c_{V}\cdot{\mathbb{E}}[W^{{\gamma}_{V}}]\cdot[\log(z)]^{k_{V}}z^{-{\gamma}_{V}}\,,\qquad\mbox{as}\quad
z\to+\infty\,;$

  * (B)

If ${\gamma}_{V}={\gamma}_{W}\eqqcolon{\gamma}$, then

${\mathbb{P}}[V\cdot
W>z]\sim\frac{{\gamma}c_{V}c_{W}}{k_{V}+1}\cdot[\log(z)]^{1+k_{V}}z^{-{\gamma}}\,,\qquad\mbox{as}\quad
z\to+\infty\,.$

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