# Asymptotic equivalence for nonparametric regression with dependent errors:
Gauss-Markov processes††thanks: This research has been supported in part by
the research grant DE 502/27-1 of the German Research Foundation (DFG).

Holger Dette  Ruhr-Universität Bochum  Martin Kroll  Ruhr-Universität Bochum

###### Abstract

For the class of Gauss-Markov processes we study the problem of asymptotic
equivalence of the nonparametric regression model with errors given by the
increments of the process and the continuous time model, where a whole path of
a sum of a deterministic signal and the Gauss-Markov process can be observed.
In particular we provide sufficient conditions such that asymptotic
equivalence of the two models holds for functions from a given class, and we
verify these for the special cases of Sobolev ellipsoids and Hölder classes
with smoothness index $>1/2$ under mild assumptions on the Gauss-Markov
process at hand. To derive these results, we develop an explicit
characterization of the reproducing kernel Hilbert space associated with the
Gauss-Markov process, that hinges on a characterization of such processes by a
property of the corresponding covariance kernel introduced by Doob [Doo49]. In
order to demonstrate that the given assumptions on the Gauss-Markov process
are in some sense sharp we also show that asymptotic equivalence fails to hold
for the special case of Brownian bridge. Our findings demonstrate that the
well-known asymptotic equivalence of the Gaussian white noise model and the
nonparametric regression model with i.i.d. standard normal errors (see Brown
and Low, [BL96]) can be extended to a result treating general Gauss-Markov
noises in a unified manner.

Keywords: Asymptotic equivalence, nonparametric regression, dependent errors,
Gauss-Markov process, triangular kernel

AMS Subject classification: 62B15, 62G08, 60G15, 62G20

##  Introduction

In a seminal paper Brown and Low [BL96] establish asymptotic equivalence of
the nonparametric regression model with discrete observations

$Y_{i,n}=f\left(\frac{i}{n}\right)+\eta_{i},\qquad i=1,\ldots,n,$ (1.1)

and the continuous time model defined by the stochastic differential equation

$\mathrm{d}Y_{t}=f(t)\mathrm{d}t+\frac{1}{\sqrt{n}}\mathrm{d}W_{t},\quad
t\in[0,1],$ (1.2)

where $\eta_{1},\ldots,\eta_{n}$ are independent, standard Gaussian random
variables, $W$ denotes a standard Brownian motion (that is, $\mathrm{d}W$ is
white noise) and $f$ is the unknown nonparametric drift satisfying a
smoothness assumption. Equation (1.2) is commonly referred to as the _Gaussian
white noise model_ and serves as an important benchmark model in nonparametric
statistics. Often, due to the absence of discretization effects, statistical
methods are easier to analyze in model (1.2) than in (1.1). Asymptotic
equivalence between the two models then suggests that theoretical results
obtained in the Gaussian white noise model hold true in the more realistic
model (1.1) as well.

Since the contribution of Brown and Low numerous authors have worked on the
problem of establishing asymptotic equivalence of various models from
different perspectives. For example, Grama and Nussbaum [GN98] investigate
nonparametric generalized linear models, and Brown and Zhang [BZ98] prove
nonequivalence when the smoothness of the function class is equal to $1/2$.
Brown et al. [Bro+02] and Reiß [Rei08] study the random design in the one-
dimensional and multivariate case, respectively (see also Carter, [Car06] for
some results in models with a multivariate fixed design). The general
framework in [BL96] is already formulated for heteroscedastic errors and
Carter [Car07] shows asymptotic equivalence for unknown variances and design
density. We also refer to the work of Reiß [Rei11] and Meister [Mei11] who
propose rate-optimal estimators of the volatility function and sharp minimax
constants in the functional linear regression model as an application of
asymptotic equivalence, respectively.

This list is by no means complete, but a common feature of most publications
in this field consists in the fact that the random variables in the
corresponding discrete nonparametric regression model are assumed to be
independent. Grama and Neumann [GN06] consider a nonparametric autoregression
model but still use an i.i.d. assumption for the innovations. Golubev,
Nussbaum, and Zhou [GNZ10] study a stationary process and show asymptotic
equivalence in the context of spectral density estimation. However, when it
comes to regression the assumption of i.i.d. errors is often made for the
theoretical analysis. Motivated by the work of Johnstone and Silverman [JS97]
and Johnstone [Joh99], Carter [Car09] considers asymptotic equivalence under
the assumption that the noise process is given by a wavelet composition with
stochastically independent coefficients. Another notable exception is the
recent work of Schmidt-Hieber [SH14], who considers the nonparametric
regression model (1.1) with fractional Gaussian noise (fGN) and establishes
the asymptotic equivalence to a model of the form (1.2), where the error
process $\frac{1}{\sqrt{n}}W_{t}$ is replaced by a fractional Brownian motion
(fBM) $n^{H-1}B_{t}^{H}$ with Hurst parameter $H\in(\frac{1}{4},1)$ over
periodic Sobolev ellipsoids containing sufficiently smooth functions. It is
also shown that asymptotic equivalence fails to hold for certain combinations
of Hurst parameter and smoothness index leading to sharp results concerning
the smoothness requirement in the case $H\in[\frac{1}{2},1)$.

The purpose of the present paper is to provide an essentially distinct way of
investigating nonparametric regression models with dependent errors for
asymptotic equivalence. Instead of replacing the Brownian motion by a
fractional Brownian motion, we consider here arbitrary Gauss-Markov processes
as error processes in model (1.2). To be precise let
$\Xi=(\Xi_{t})_{t\in[0,1]}$ be such a Gauss-Markov process with initial state
$\Xi_{0}=0$, and consider the continuous time model

$Y_{t}=\int_{0}^{t}f(s)\mathrm{d}s+\frac{1}{\sqrt{n}}\Xi_{t},\quad t\in[0,1].$
(1.3)

Assuming equidistant design points, the candidate regression model for
asymptotic equivalence is given by

$Y_{i,n}=f\left(\frac{i}{n}\right)+\sqrt{n}\xi_{i,n},\quad i=1,\ldots,n,$
(1.4)

where
$\xi_{i,n}\stackrel{{\scriptstyle\mathcal{L}}}{{=}}\Xi_{i/n}-\Xi_{(i-1)/n}$
denote the increments of the process $\Xi$. Note that in general the
observation errors $\xi_{i,n}$ are not uncorrelated. However, for the special
case of Brownian motion, that is
$\Xi\stackrel{{\scriptstyle\mathcal{L}}}{{=}}W$, Equations (1.1) and (1.4) are
equivalent in distribution since Brownian motion has independent increments
and satisfies the scaling property that $W(\boldsymbol{\cdot})$ and
$c^{-1/2}W(c\,\boldsymbol{\cdot})$ have the same distribution for any $c>0$.
The main results of the present paper establish the asymptotic equivalence of
the models (1.3) and (1.4) for a wide class of Gauss-Markov processes for
functional parameters $f$ belonging to a Sobolev or Hölder class of
sufficiently smooth functions.

There are different ways to prove asymptotic equivalence between regression
and white noise experiments in the literature. The original paper of Brown and
Low [BL96] considers the case where the regression $f$ is an element of a
function class, say $\Theta$, and uses the key assumption

$\lim\limits_{n\to\infty}\sup_{f\in\Theta}n\int_{0}^{1}\Big{(}f(t)-\bar{f}_{n}(t)\Big{)}^{2}\mathrm{d}t=0,$
(1.5)

where
$\bar{f}_{n}(\boldsymbol{\cdot})=\sum_{j=1}^{n}f(j/n)\mathbf{1}_{((j-1)/n,j/n]}(\boldsymbol{\cdot})$
denotes a piecewise constant approximation of $f$ and $\mathbf{1}_{A}$ is the
indicator function of the set $A$. These authors introduce an intermediate set
of random variables that forms a sufficient statistic for the white noise
model with the function $f$ replaced with $\bar{f}_{n}$. Then, the Hellinger
distance between this sufficient statistic and the regression experiment with
discrete sampling locations is shown to converge to zero. On the other hand,
condition (1.5) guarantees that the Le Cam distance between the white noise
models with parameters $f$ and $\bar{f}_{n}$, respectively, vanishes as
$n\to\infty$. Note that this approach can be used to show asymptotic
equivalence but cannot provide the optimal rate of convergence for the Le Cam
distance between the two experiments. This issue has been solved by Rohde
[Roh04], who considers a Gaussian sequence space model as an intermediate
experiment between the two experiments of interest. Schmidt-Hieber [SH14] in
some sense generalizes the conditions in [BL96] and formalizes them in the
framework of reproducing kernel Hilbert spaces (RKHSs), which is a suitable
setup for the investigation in the case of fractional Brownian motion. Our
analysis of the asymptotic equivalence for models with Gauss-Markov errors
will also be based on the RKHS framework. As an essential ingredient we will
use a characterization of Gauss-Markov processes introduced by Doob [Doo49] to
derive an explicit representation of the RKHSs associated with the Gauss-
Markov processes under consideration, which can be used to develop sufficient
conditions for the asymptotic equivalence of the models (1.3) and (1.4).

The remaining part of the paper is organized as follows. In Section 2 we
introduce Gauss-Markov processes and recap the characterization of such
processes introduced by Doob [Doo49] that will be pivotal for our approach.
Roughly speaking, these processes can be characterized by the property that
the corresponding covariance kernel is triangular (see Section 2.1 for a
precise definition). In Section 2.2 we study the reproducing kernel Hilbert
spaces (RKHSs) associated with Gauss Markov processes and derive
representations of these spaces via Hilbert space isomorphisms into a space of
square-integrable functions. In Section 3 we recall the basic elements of Le
Cam theory and provide sufficient conditions on the class of potential
functions $f$ and the Gauss-Markov process that imply asymptotic equivalence.
The characterizations of the RKHSs are used in Section 4 where we establish
asymptotic equivalence of the models (1.3) and (1.4) under mild assumptions on
the Gauss-Markov process in model (1.3) for Sobolev ellipsoids and Hölder
classes. In Section 4.3 we discuss a different approach to derive such
results, which is not based on RKHS theory. It requires a (slightly) stronger
assumption than those used in this paper and applies results from the seminal
paper [BL96] to suitable transformations of the experiments of interest.

Finally, in Section 4.4 we demonstrate that asymptotic equivalence cannot hold
without any additional assumptions on the Gauss-Markov process. More
precisely, for the special case of Brownian bridge we show that the Le Cam
distance between the two experiments is bounded away from zero. The proofs of
our results are deferred to the Appendix.

### Notation

Vectors will be denoted with bold letters (i.e., we write
$\boldsymbol{x}_{n}=(x_{1,n},\ldots,x_{m(n),n})$ when both length and entries
of a vector might vary with $n$). We also use the shorthand
$\boldsymbol{x}_{1:k,n}=(x_{1,n},\ldots,x_{k,n})$. We write $a_{n}\lesssim
b_{n}$ if $a_{n}\leq Cb_{n}$ for a constant $C$ that is independent of $n$.
The shorthand $a_{n}\asymp b_{n}$ is used when $a_{n}\lesssim b_{n}$ and
$b_{n}\lesssim a_{n}$ hold simultaneously.

##  The RKHS associated with a Gauss-Markov process

The purpose of this section is to lay the foundations for our main results in
Sections 3 and 4. In Section 2.1 we recapitulate a characterization of Gauss-
Markov process going back to Doob which will be important for our further
reasoning. As a consequence of this characterization we can give an explicit
description of the RKHS associated with the covariance kernel of a Gauss-
Markov process, which is of independent interest and presented in Section 2.2.

###  Gauss-Markov processes

By definition a Gauss-Markov process is a stochastic process that is both
Gaussian and Markov. Such a process $X=(X_{t})_{t\in[0,1]}$ is essentially
characterized by the following factorization property of the covariance
function:

$K_{X}(s,t)\vcentcolon=\operatorname{Cov}(X_{s},X_{t})=\mathbf{E}[X_{s}X_{t}]=U(s)V(t)\quad\text{for
}0\leq s\leq t\leq 1,$ (2.1)

where $U$ and $V$ are (known) non-negative functions on the interval $[0,1]$;
see [Doo49, MM65] for details. Kernels with the factorization property (2.1)
are sometimes referred to as _triangular kernels_ in the literature. Examples
of Gauss-Markov processes include standard Brownian motion ($U(t)=t$, $V\equiv
1$), the Ornstein-Uhlenbeck process ($U(t)=\exp(Lt)$ and $V(t)=\exp(-Lt)$ for
some $L>0$) and the Brownian bridge ($U(t)=t$, $V(t)=1-t$).

For our results, we will further assume that the considered Gauss-Markov
process starts in zero. Such a process will be denoted with
$\Xi=(\Xi_{t})_{t\in[0,1]}$ from now on, and we impose the following
assumption.

######  Assumption 2.1.

The process $\Xi=(\Xi_{t})_{t\in[0,1]}$ is a Gauss-Markov process with
$\Xi_{0}=0$, and non-degenerate on the open interval $(0,1)$.

Note that Assumption 2.1 implies that there exist functions $u$ and $v$ in the
representation

$K_{\Xi}(s,t)=u(s)v(t)$ (2.2)

of the covariance kernel satisfying
$u(\boldsymbol{\cdot})v(\boldsymbol{\cdot})\geq 0$ on the interval $[0,1]$,
$u(\boldsymbol{\cdot})v(\boldsymbol{\cdot})>0$ on $(0,1)$ and that the
function

$q(t)=\frac{u(t)}{v(t)},$

is continuous on the interval $[0,1)$, non-negative and strictly increasing on
$[0,1]$ (see [MM65], p. 507). Moreover, under Assumption 2.1, the Gauss-Markov
process $\Xi$ can be written in distribution as

$\Xi_{t}=v(t)\cdot W_{q(t)}.$ (2.3)

Vice versa, this transformation of Brownian motion defines a Gaussian process
with covariance kernel (2.2). Note that a simple calculation shows that the
process $\Xi$ has independent increments if and only if the function $v$ is
constant.

Let us shortly explain how processes $\Xi$ satisfying Assumption 2.1 can be
obtained. First, one can of course define such a process directly by means of
the representation (2.3) provided that the function $u$ and $v$ satisfy the
properties stated above and $u(0)=0$ or $v(0)=0$ (this latter assumption
guarantees that the process starts in $0$). Second, starting with a general
centered non-degenerate Gauss-Markov process $X=(X_{t})_{t\in[0,1]}$ with
covariance kernel (2.1), and then conditioning on $X_{0}=0$ does neither
suspend Gaussianity nor the Markov property. More precisely, the process
$\Xi=(\Xi_{t})_{t\geq 0}$ with $\Xi_{t}\sim X_{t}|\\{X_{0}=0\\}$ is a centered
Gaussian process with covariance kernel

$K_{\Xi}(s,t)\vcentcolon=\operatorname{Cov}(\Xi_{s},\Xi_{t})=\mathbf{E}[\Xi_{s}\Xi_{t}]=u(s)v(t),\qquad
0\leq s\leq t\leq 1,$

where the functions $u$ and $v$ are related to $U$ and $V$ through the
identities

$u(t)=U(t)-Q(0)V(t)\qquad\text{and}\qquad v(t)=V(t),$

where $Q(t)=U(t)/V(t)$. In particular, the process $\Xi$ satisfies Assumption
2.1, and $q(t)=u(t)/v(t)$ can be represented as $q(t)=Q(t)-Q(0)$.

Surprisingly, Gauss-Markov processes starting at zero can also be obtained by
conditioning a non-Markovian Gaussian process $Y=(Y_{t})_{t\geq 0}$ on the
event $\\{Y_{0}=0\\}$. Then, the initial Gaussian process $Y$ is called
_conditionally Markov_. As one interesting example (further examples of
conditionally Markov processes can again be found in [MM65], pp. 513 and 516)
let us state the stationary Gaussian process $(Y_{t})_{t\in\mathbb{R}}$
defined on the whole real-line with zero mean and covariance kernel

$\displaystyle K_{Y}(x,y)=\begin{cases}1-\lvert x-y\rvert,&\text{ if
}0\leq\lvert x-y\rvert\leq 1,\\\ 0,&\text{ if }\lvert x-y\rvert>1.\end{cases}$

for $x,y\in\mathbb{R}$. Then, the process $(\Xi_{t})_{t\geq 0}$ with
$\Xi_{t}\sim Y_{t}|\\{Y_{0}=0\\}$ is a centered Gaussian process with
covariance kernel

$K_{\Xi}(t_{1},t_{2})=\begin{cases}t_{1}(2-t_{2}),&\text{ if }0\leq t_{1}\leq
t_{2}\leq 1,\\\ 1-(t_{2}-t_{1}),&\text{ if }0<t_{2}-t_{1}<1,t_{2}\geq
1,t_{1}\geq 0,\\\ 0,&\text{ if }t_{2}-t_{1}\geq 0,t_{1}\geq 0.\end{cases}$

In particular, the restriction of $\Xi$ on the interval $[0,1]$ is a Gauss-
Markov process satisfying Assumption 2.1, and $K_{\Xi}(s,t)=u(s)v(t)$ for
$0\leq s\leq t\leq 1$ with $u(x)=x$ and $v(x)=2-x$. This process has been
considered by Slepian in [Sle61] where it was proved that the process fulfills
a ’peculiar Markov-like property’. The restriction of the process $\Xi$ on the
unit interval fulfills all the technical assumptions made on the functions $u$
and $v$ below, and thus the results on asymptotic equivalence derived in
Section 3 and 4 hold in particular for this process.

###  Reproducing kernel Hilbert spaces

We briefly recall some basic facts about reproducing kernel Hilbert spaces.
For a detailed discussion the reader is referred to the monographs of Berlinet
and Thomas-Agnan [BTA04] and Paulsen and Raghupathi [PR16]. A subset
$\mathcal{H}$ of the set $\mathcal{F}(\mathcal{X},\mathbb{R})$ of real-valued
functions on a domain $\mathcal{X}$ is called a reproducing kernel Hilbert
space (RKHS) on $\mathcal{X}$ if

  * $\bullet$

$\mathcal{H}$ is a vector subspace of $\mathcal{F}(\mathcal{X},\mathbb{R})$,

  * $\bullet$

$\mathcal{H}$ has an inner product
$\langle\boldsymbol{\cdot},\boldsymbol{\cdot}\rangle_{\mathcal{H}}$ with
respect to which $\mathcal{H}$ is a Hilbert space,

  * $\bullet$

for any $x\in\mathcal{X}$, the linear evaluation functional
$\ell_{x}\colon\mathcal{H}\to\mathbb{R}$ with $\ell_{x}(f)=f(x)$ is bounded.

The Riesz representation theorem implies that for any $x\in\mathcal{X}$ there
exists a unique function $k_{x}\in\mathcal{H}$ such that for every
$f\in\mathcal{H}$ one has $f(x)=\ell_{x}(f)=\langle
f,k_{x}\rangle_{\mathcal{H}}$. The function
$K\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}$

$K(x,y)=k_{x}(y)$

is called the _reproducing kernel_ for $\mathcal{H}$. Reproducing kernels are
_positive definite_ in the sense that the inequality

$\sum_{i,j=1}^{n}a_{i}\overline{a}_{j}K(x_{i},x_{j})\geq 0$

holds for all $n\in\mathbb{N}$, $x_{i}\in\mathcal{X}$ and
$a_{i}\in\mathbb{C}$. Vice versa, for any positive definite function $K$ on
$\mathcal{X}$ there exists a RKHS $\mathcal{H}(K)$ on $\mathcal{X}$ with
reproducing kernel $K$ and this RKHS is uniquely determined by the kernel. In
particular, this holds true for covariance kernels. Finally, it is well-known
that the linear span of the functions $k_{y}=K(\boldsymbol{\cdot},y)$ is dense
in $\mathcal{H}$.

In the following, we will consider the RKHS $\mathcal{H}(\Xi)$ associated with
the covariance kernel $K_{\Xi}\colon[0,1]\times[0,1]\to\mathbb{R}$ of a Gauss-
Markov process $\Xi$ satisfying Assumption 2.1 as introduced in Section 2. As
a consequence

$K_{\Xi}(s,t)=u(s)v(t),\qquad 0\leq s\leq t\leq 1.$

Let us from now on assume that $q(0)=0$ for $q(x)=u(x)/v(x)$ which is under
the validity of Assumption 2.1 certainly satisfied whenever $v(0)\neq 0$. This
additional condition is valid for all the results in Section 4. As mentioned
above, $\Xi$ can be represented in distribution as

$\Xi_{t}=v(t)W_{q(t)}$

where $W$ is standard Brownian motion. Denote $T=q(1)$, and let us consider
the mapping $\psi\colon\mathcal{H}(\Xi)\to L^{2}([0,T])$ defined on the
’generators’ $\\{K_{\Xi}(\boldsymbol{\cdot},t)$, $t\in[0,1]\\}$ via

$K_{\Xi}(\boldsymbol{\cdot},t)\stackrel{{\scriptstyle\psi}}{{\mapsto}}v(t)\mathbf{1}_{[0,q(t)]}(\boldsymbol{\cdot})$
(2.4)

(on general elements of $\mathcal{H}(\Xi)$ is naturally defined by a limiting
process). The mapping $\psi$ is an isometry of Hilbert spaces since

$\displaystyle\langle
K_{\Xi}(\boldsymbol{\cdot},s),K_{\Xi}(\boldsymbol{\cdot},t)\rangle_{\mathcal{H}(\Xi)}$
$\displaystyle=K_{\Xi}(s,t)=v(s)v(t)q(s)$
$\displaystyle=\int_{0}^{T}v(s)v(t)\mathbf{1}_{[0,q(s)]}(u)\mathbf{1}_{[0,q(t)]}(u)\mathrm{d}u$
$\displaystyle=\langle\psi K_{\Xi}(\boldsymbol{\cdot},s),\psi
K_{\Xi}(\boldsymbol{\cdot},t),\rangle_{L^{2}([0,T])}$

for $0\leq s\leq t\leq 1$ (this holds true due to the strict monotonicity and
continuity of $q$ combined with the fact that the indicator functions
$\mathbf{1}_{[0,t]},t\in[0,T]$, are dense in $L^{2}([0,T])$). We use the
shorthand notation
$\psi_{t}=\psi(K_{\Xi}(\boldsymbol{\boldsymbol{\cdot}},t))$. For
$F\in\mathcal{H}(\Xi)$ we obtain with $g=\psi F\in L^{2}([0,T])$ the
representation

$\displaystyle F(t)$ $\displaystyle=\langle
F,K_{\Xi}(\boldsymbol{\cdot},t)\rangle_{\mathcal{H}(\Xi)}=\langle\psi
F,\psi_{t}\rangle_{L^{2}([0,T])}=\langle g,\psi_{t}\rangle_{L^{2}([0,T])}$
$\displaystyle=\int_{0}^{T}g(u)v(t)1_{[0,q(t)]}(u)\mathrm{d}u=v(t)\int_{0}^{q(t)}g(u)\mathrm{d}u\leavevmode\nobreak\
.$

This gives an explicit characterization of the RKHS $\mathcal{H}(\Xi)$, namely

$\mathcal{H}(\Xi)=\left\\{F:[0,1]\to\mathbb{R}\leavevmode\nobreak\
\Big{|}\leavevmode\nobreak\ F(t)=v(t)\int_{0}^{q(t)}g(u)\mathrm{d}u\text{ for
some }g\in L^{2}([0,T])\right\\},$ (2.5)

and this space is equipped with the norm $\lVert
F\rVert_{\mathcal{H}(\Xi)}=\lVert g\rVert_{L^{2}([0,T])}$ for $F$ as in (2.5).
Of course, in the special case of Brownian motion ($u(t)=t,v\equiv 1$) these
calculations yield the well-known fact that the RKHS corresponding to the
kernel of standard BM contains exactly the primitives (starting at $0$) of
square-integrable functions. If
$F(\boldsymbol{\cdot})=F_{f}(\boldsymbol{\cdot})=\int_{0}^{\boldsymbol{\cdot}}f(s)\mathrm{d}s$,
then $f$ can be derived by differentiation:

$\displaystyle f(t)$ $\displaystyle=\frac{\partial}{\partial
t}\left[v(t)\int_{0}^{q(t)}g(u)\mathrm{d}u\right]$
$\displaystyle=v^{\prime}(t)\int_{0}^{q(t)}g(u)\mathrm{d}u+v(t)g(q(t))q^{\prime}(t).$

Moreover, the function $g$ can be obtained from $F_{f}$ (and thus from $f$) by

$\displaystyle g(q(t))$
$\displaystyle=\left(\frac{F_{f}(t)}{v(t)}\right)^{\prime}\cdot\frac{1}{q^{\prime}(t)}.$
(2.6)

This relation between $f$ and $g$ will be exploited when proving asymptotic
equivalence results for concrete function classes in Section 4.

##  Abstract conditions for asymptotic equivalence

In this section, we derive sufficient conditions for an abstract function
class $\Theta$ that guarantee asymptotic equivalence of nonparametric
regression with dependent errors and a continuous time model with an additive
error process that is Gauss-Markov. In Section 3.1 we gather the necessary
notions from asymptotic equivalence theory. In Section 3.2, we state the main
result that provides the announced sufficient conditions for asymptotic
equivalence of models (1.3) and (1.4).

###  Asymptotic equivalence of experiments

Le Cam’s equivalence theory for statistical experiments has by now become a
common tool in nonparametric statistics. As one of its major appeals one might
consider the fact that complex statistical models can be shown equivalent to
simple and well-studied benchmark models, at least asymptotically when the
amount of information contained in the data increases. This is not only of
interest in its own but has also turned out useful when proving optimality
properties of estimation techniques. For instance, Reiß [Rei11] proposes rate-
optimal estimators of the volatility function and simple efficient estimators
of the integrated volatility as an application of Le Cam theory. Similarly,
Meister [Mei11] derives sharp minimax constants in the functional linear
regression model as an application of asymptotic equivalence of this model and
an inverse problem in a white noise setup. The interested reader will find
comprehensive introductions into Le Cam theory in the monographs [LC86, LCY00,
Tor91] as well as in the recent survey paper [Mar16] by Mariucci, the latter
focusing on nonparametric models that are also the topic of the present paper.

There exist various equivalent ways to introduce the concept of equivalence of
statistical experiments. One approach in the general theory introduced by Le
Cam is by means of the abstract concept of _transitions_. In the case of
dominated statistical models with Polish sample spaces (which is exclusively
considered in this article), however, this general notion essentially boils
down to the class of _Markov kernels_ as has been shown in Proposition 9.2 in
[Nus96]. To be precise, let
$\mathfrak{E}_{i}=(\mathcal{X}_{i},\mathscr{X}_{i},(\mathbf{P}_{i,\theta})_{\theta\in\Theta})$
for $i\in\\{1,2\\}$ be two such dominated experiments with Polish parameter
spaces $(\mathcal{X}_{i},\mathscr{X}_{i})$, and sharing the same parameter
space $\Theta$. Then, the _deficiency_ of $\mathfrak{E}_{1}$ with respect to
$\mathfrak{E}_{2}$ is defined as the quantity

$\delta(\mathfrak{E}_{1},\mathfrak{E}_{2})=\inf_{K}\sup_{\theta\in\Theta}\lVert
K\mathbf{P}_{1,\theta}-\mathbf{P}_{2,\theta}\rVert_{\mathrm{TV}}\leavevmode\nobreak\
,$

where $\lVert\mathbf{P}-\mathbf{Q}\rVert_{\mathrm{TV}}$ denotes the total
variation distance between probability measures $\mathbf{P}$ and $\mathbf{Q}$,
and the infimum is taken over all Markov kernels
$K\colon\mathcal{X}_{1}\times\mathscr{X}_{2}\to[0,1]$.
$\delta(\mathfrak{E}_{1},\mathfrak{E}_{2})=0$ has the interpretation that the
experiment $\mathfrak{E}_{1}$ is more informative than $\mathfrak{E}_{2}$.

Building on the definition of deficiency, the _Le Cam distance_ between
$\mathfrak{E}_{1}$ and $\mathfrak{E}_{2}$ is defined by symmetrization as

$\Delta(\mathfrak{E}_{1},\mathfrak{E}_{2})=\max\\{\delta(\mathfrak{E}_{1},\mathfrak{E}_{2}),\delta(\mathfrak{E}_{2},\mathfrak{E}_{1})\\},$

which provides a pseudo-metric on the space of all statistical models with
common parameter space. Two experiments $\mathfrak{E}_{1}$ and
$\mathfrak{E}_{2}$ are called _equivalent_ if
$\Delta(\mathfrak{E}_{1},\mathfrak{E}_{2})=0$. More generally, two sequences
$(\mathfrak{E}_{1,n})_{n\in\mathbb{N}}$ and
$(\mathfrak{E}_{2,n})_{n\in\mathbb{N}}$ are said to be _asymptotically
equivalent_ if $\Delta(\mathfrak{E}_{1,n},\mathfrak{E}_{2,n})\to 0$. In the
latter case we write $\mathfrak{E}_{1,n}\approx\mathfrak{E}_{2,n}$.

###  Sufficient conditions for asymptotic equivalence

We now rigorously define the two statistical experiments that will be examined
for asymptotic equivalence in this paper. Let $\Theta$ denote a given class of
functions. The first experiment with discrete observations is given by

$\mathfrak{E}_{1,n}(\Theta)=(\mathbb{R}^{n},\mathscr{B}(\mathbb{R}^{n}),(\mathbf{P}_{1,n}^{f})_{f\in\Theta})\leavevmode\nobreak\
,$

where $\mathbf{P}_{1,n}^{f}$ denotes the distribution of the vector
$\mathbf{Y}_{n}=(Y_{1,n},\ldots,Y_{n,n})$ with components $Y_{i,n}$ defined by

$Y_{i,n}=f(t_{i,n})+\sqrt{n}\xi_{i,n},\quad i=1,\ldots,n,$ (3.1)

where $t_{i,n}=i/n$ are the sampling locations, and
$\xi_{i,n}=\Xi\left(t_{i,n}\right)-\Xi\left(t_{i-1,n}\right)$ are the
increments of a centered Gauss-Markov process $\Xi=(\Xi_{t})_{t\in[0,1]}$ with
$\Xi_{0}=0$ as introduced in Section 2.

The second experiment with continuous observations is given through

$\mathfrak{E}_{2,n}(\Theta)=(\mathcal{C}([0,1],\mathbb{R}),\mathscr{B}(\mathcal{C}([0,1],\mathbb{R})),(\mathbf{P}_{2,n}^{f})_{f\in\Theta})\leavevmode\nobreak\
,$

where $\mathbf{P}_{2,n}^{f}$ denotes the distribution of the continuous time
process $(Y_{t})_{t\in[0,1]}$ defined by

$Y_{t}=\int_{0}^{t}f(s)\mathrm{d}s+\frac{1}{\sqrt{n}}\Xi_{t},\quad t\in[0,1].$
(3.2)

Typical realizations from the two experiments are visualized in Figure 1 for
the case of an Ornstein-Uhlenbeck process conditioned to start in zero.

Figure 1: Exemplary realizations of the observations in the two experiments
$\mathfrak{E}_{1,n}$ (left) and $\mathfrak{E}_{2,n}$ (right) for $n=50$ where
the Gauss-Markov process is given through the functions $u(x)=\exp(x)$ and
$q(x)=\exp(2x)-1$. In both figures the actual observations are plotted in
blue. The black dotted line represents the true function $f$ in the left plot,
and its anti-derivative
$F_{f}(\boldsymbol{\cdot})=\int_{0}^{\boldsymbol{\cdot}}f(s)\mathrm{d}s$ in
the right plot. The error terms, that is, the increments of $\Xi$ for
$\mathfrak{E}_{1,n}$ and the process $\Xi$ itself for $\mathfrak{E}_{2,n}$,
are plotted in red (note that there is a secondary y-axis in red for the error
terms).

The following result provides sufficient conditions for asymptotic equivalence
of the experiments $\mathfrak{E}_{1,n}$ and $\mathfrak{E}_{2,n}$ in terms of
the class $\Theta$ and the Gauss-Markov process $\Xi$. The proof is given in
Appendix A.

######  Theorem 3.1.

Let $\Xi$ be a Gauss-Markov process on the interval $[0,1]$ with $\Xi_{0}=0$
such that Assumption 2.1 holds with $K_{\Xi}(s,t)=u(s)v(t)$ for $0\leq s\leq
t\leq 1$, and $v(1)\neq 0$. Set
$F_{f}(\boldsymbol{\boldsymbol{\cdot}})=\int_{0}^{\boldsymbol{\boldsymbol{\cdot}}}f(s)\mathrm{d}s$
and assume that the conditions

  1. (i)

$\frac{1}{n}\sup_{f\in\Theta}\sum_{i=1}^{n}\frac{(f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s)^{2}}{v^{2}(t_{i,n})(q(t_{i,n})-q(t_{i-1,n}))}\to
0$ and

  2. (ii)

$\sqrt{n}\sup_{f\in\Theta}\inf_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\lVert
F_{f}(\boldsymbol{\cdot})-\sum_{i=1}^{n}\alpha_{i}K_{\Xi}\left(\boldsymbol{\cdot},t_{i,n}\right)\rVert_{\mathcal{H}(\Xi)}\to
0$

are satisfied, where $t_{i,n}=i/n$ and
$\lVert\,\boldsymbol{\cdot}\,\rVert_{\mathcal{H}(\Xi)}$ is the norm in the
RKHS $\mathcal{H}(\Xi)$. Then, the two sequences of experiments
$(\mathfrak{E}_{1,n}(\Theta))_{n\in\mathbb{N}}$ and
$(\mathfrak{E}_{2,n}(\Theta))_{n\in\mathbb{N}}$ are asymptotically equivalent,
i.e.

$\mathfrak{E}_{1,n}(\Theta)\approx\mathfrak{E}_{2,n}(\Theta).$

##  Specific function classes and a counterexample

In this section, we first apply the abstract Theorem 3.1 in order to establish
asymptotic equivalence over Sobolev ellipsoids (Section 4.1) and Hölder balls
(Section 4.2) of sufficiently smooth functions. In Section 4.3 we discuss an
alternative proof strategy, which is based on some transformations of the
different experiments and requires stronger assumptions as used in Section 4.1
and 4.2. Finally, in Section 4.4 we give a counterexample which shows that
asymptotic equivalence cannot hold (even for function classes containing very
smooth functions) in general if only Assumption 2.1 is required.

###  Asymptotic equivalence for Sobolev ellipsoids

Consider the (complex) trigonometric basis
$\\{\boldsymbol{\mathrm{e}}_{k}\\}_{k\in\mathbb{Z}}$ of $L^{2}([0,1])$ given
by

$\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot})=\exp(-2\pi\mathrm{i}k\boldsymbol{\cdot}).$

Any function $f$ in $L^{2}([0,1])$ can be represented as a convergent series
(in $L^{2}$-sense) of the form

$f(\boldsymbol{\cdot})=\sum_{k=-\infty}^{\infty}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot})$

where $\theta_{k}=\langle
f,\boldsymbol{\mathrm{e}}_{k}\rangle_{L^{2}}=\int_{0}^{1}f(s)\exp(2\pi\mathrm{i}ks)\mathrm{d}s$
are the uniquely defined Fourier coefficients of the function $f$. As
potential function classes over which we would like to establish asymptotic
equivalence we consider Sobolev ellipsoids

$\Theta(\beta,L)=\left\\{f(\boldsymbol{\cdot})=\sum_{k=-\infty}^{\infty}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot}):\theta_{k}=\overline{\theta_{-k}}\text{
and }\sum_{k=-\infty}^{\infty}(1+\lvert
k\rvert)^{2\beta}\lvert\theta_{k}\rvert^{2}\leq L^{2}\right\\}.$ (4.1)

Note that the condition that $\theta_{k}=\overline{\theta_{-k}}$ is used only
to ensure that the functions in the set $\Theta(\beta,L)$ are real-valued. The
following result establishes asymptotic equivalence of the experiments
$\mathfrak{E}_{1,n}(\Theta(\beta,L))$ and
$\mathfrak{E}_{2,n}(\Theta(\beta,L))$ considered in Section 3 for
$\beta>{1}/{2}$ under some assumptions on the functions $v$ and $q$
characterizing the Gauss-Markov process $(\Xi_{t})_{t\in[0,1]}$.

######  Theorem 4.1.

Let $\Xi$ be a Gauss-Markov process on the interval $[0,1]$ with $\Xi_{0}=0$
such that Assumption 2.1 holds with $K_{\Xi}(s,t)=u(s)v(t)$ for $0\leq s\leq
t\leq 1$. If $\beta>1/2$ and the assumptions

  1. a)

$q\in C^{1}$ and
$0<\inf_{t\in[0,1]}q^{\prime}(t)\leq\sup_{t\in[0,1]}q^{\prime}(t)\leq
C<\infty$,

  2. b)

$\inf_{t\in[0,1]}v(t)>0$, and

  3. c)

$v^{\prime}$ and $q^{\prime}$ are Hölder continuous with index $\gamma>1/2$

are satisfied, then

$\mathfrak{E}_{1,n}(\Theta(\beta,L))\approx\mathfrak{E}_{2,n}(\Theta(\beta,L)).$

The proof of Theorem 4.1 is complicated and deferred to Appendix B.1. We
proceed with some remarks that put the statement in the context of well-known
results for the Brownian motion.

######  Remark 4.2.

It is known that for the smoothness index $\beta=1/2$ asymptotic equivalence
between nonparametric regression model (1.1) and the white noise with drift
experiment (1.2) (i.e., our experiment $\mathfrak{E}_{2,n}$ for the special
case of Brownian motion) does not hold. See Remark 4.6 in [BL96] for further
details. In Section 4.4 below we consider the example of a Brownian bridge as
an error process and prove that asymptotic equivalence for arbitrary values of
$\beta$ cannot hold without additional assumptions on the Gauss-Markov process
(like, for instance, assumptions on the functions $v$ and $q$ as made in
Theorem 4.1).

######  Remark 4.3.

In the proof of Theorem 4.1 it is shown that under our assumptions the Le Cam
distance between the experiments $\mathfrak{E}_{1,n}(\Theta(\beta,L))$ and
$\mathfrak{E}_{2,n}(\Theta(\beta,L))$ converges to zero with a rate that
cannot be faster than

$\mathcal{O}(\max\\{n^{-1/2},n^{-\beta+1/2}\\})$

(indeed, the rate might even be slower due to the terms incorporating the
Hölder index $\gamma$ from Condition c in the statement of the theorem). In
the case where the Gauss-Markov is standard Brownian motion the rate
$n^{-\beta+1/2}$ can be established which is faster for $\beta>1$; see Rohde,
[Roh04] for further details. Since we have restricted ourselves to proving
asymptotic equivalence, we have not further pursued this issue here.

######  Remark 4.4 (Sequence space representations).

Note that for the special case of Brownian motion the relation between the
functions $f=F^{\prime}$ and $g$ in in the representation (2.5) of
$F\in\mathcal{H}(\Xi)$ reduces to the equality $g=f$. Hence, by representing
$f$ in terms of the coefficients of a series expansion with respect to an
orthonormal basis (for instance, the complex trigonometric basis given through
$\\{\boldsymbol{\mathrm{e}}_{k}\\}_{k\in\mathbb{Z}}$) the same representation
is achieved for $g$. Via the isomorphism $\psi$ defined in (2.4) (more
precisely, its inverse $\psi^{-1}$) this yields a representation of $F_{f}$ in
terms of an orthonormal basis in the RKHS leading to a Gaussian sequence space
model of the form

$y_{k}=\theta_{k}+\xi_{k},\qquad k\in\mathbb{Z},$

where $\xi_{k}$ i.i.d. $\sim\mathcal{N}(0,1)$ and the sequence
$\theta=(\theta_{k})_{k\in\mathbb{Z}}$ satisfies the same smoothness
conditions as in the definition of the Sobolev ellipsoid $\Theta(\beta,L)$. In
the general case of an arbitrary Gauss-Markov process the relation (2.6) does
not ’transport’ an orthonormal basis for the representation of $f$ to an
orthonormal basis for $g$.

An alternative sequence space representation (for general Gaussian processes)
is obtained from (3.2) using the Karhunen-Loeve representation of the process
$\Xi$. Note that the kernel $K_{\Xi}$ gives rise to an integral operator
defined through

$(K_{\Xi}f)(s)=\int_{0}^{1}K_{\Xi}(s,t)f(t)\mathrm{d}t$

for $f\in L^{2}([0,1])$ (by slight abuse of notation, we use the same letter
for both the kernel and the operator associated with it). Provided that (the
kernel) $K_{\Xi}$ is continuous, Mercer’s theorem applies and (the operator)
$K_{\Xi}$ has a countable system of eigenfunctions $\varphi_{k}$ with
corresponding eigenvalues $\lambda_{k}\geq 0$. By Proposition 3.11 in Neveu
[Nev68] an orthonormal basis of the RKHS $\mathcal{H}(\Xi)$ is given by the
functions $g_{k}=\sqrt{\lambda_{k}}\varphi_{k}$, $k=1,2,\ldots$ with
$\lambda_{k}>0$. In terms of this basis, (3.2) can be rewritten as

$y_{k}=\theta_{k}+\frac{1}{\sqrt{n}}\eta_{k},\quad k=1,2,\ldots$ (4.2)

where $\theta_{1},\theta_{2},\ldots$ are the Fourier coefficients of the
function $F$ with respect to the basis $g_{1},g_{2},\ldots$ and
$\eta_{1},\eta_{2},\ldots$ are independent and standard normal distributed
(note that the sequence model here is stated in terms of the coefficients of
$F_{f}$ instead of those of $f$ itself). In the special case of Brownian
motion, one has (see Beder, [Bed87], p. 66)

$g_{k}(t)=\frac{\sqrt{2}}{(k-\frac{1}{2})\pi}\sin\left((k-\frac{1}{2})\pi
t\right)=\sqrt{\lambda_{k}}\varphi_{k},\quad\lambda_{k}=\frac{1}{(k-\frac{1}{2})^{2}\pi^{2}}.$

In this case, the derivatives $\psi_{k}=g_{k}^{\prime}$ of the basis functions
$g_{k}$ in the RKHS form an orthonormal basis of $L^{2}([0,1])$. Hence, the
sequence space model (4.2) can be interpreted both as a perturbation of the
coefficients of $F_{f}$ or of $f$ (note that $f\in L^{2}([0,1])$ if and only
if $F_{f}\in\mathcal{H}(\Xi)$).

###  Asymptotic equivalence for Hölder classes

For a smoothness index $0<\alpha\leq 1$ and constants $0<L<\infty$,
$0<M\leq\infty$, we introduce the Hölder class

$\mathcal{F}(\alpha,L,M)=\left\\{f\colon[0,1]\to\mathbb{R}:\sup_{0\leq x<y\leq
1}\frac{\lvert f(x)-f(y)\rvert}{\lvert x-y\rvert^{\alpha}}\leq L\text{ and
}\lVert f\rVert_{\infty}\leq M\right\\}\leavevmode\nobreak\ ,$

Where the case $M=\infty$ means that the assumption $\lVert
f\rVert_{\infty}\leq M$ is omitted in the definition of
$\mathcal{F}(\alpha,L,M)$. Under the same technical assumptions on the
functions $v$ and $q$ as stated in Theorem 4.1 we obtain asymptotic
equivalence for functions from Hölder classes with smoothness index
$\alpha>1/2$.

######  Theorem 4.5.

Let $\Xi$ be a Gauss-Markov process on the interval $[0,1]$ with $\Xi_{0}=0$
such that Assumption 2.1 holds with $K_{\Xi}(s,t)=u(s)v(t)$ for $0\leq s\leq
t\leq 1$. Suppose further that $\alpha>1/2$ and that the assumptions

  1. a)

$q\in C^{1}$ and
$0<\inf_{t\in[0,1]}q^{\prime}(t)\leq\sup_{t\in[0,1]}q^{\prime}(t)\leq
C<\infty$,

  2. b)

$\inf_{t\in[0,1]}v(t)>0$, and

  3. c)

$v^{\prime}$ and $q^{\prime}$ are Hölder continuous with index $\gamma>1/2$

are satisfied. Then, if $M<\infty$, we have

$\mathfrak{E}_{1,n}(\mathcal{F}(\alpha,L,M))\approx\mathfrak{E}_{2,n}(\mathcal{F}(\alpha,L,M)).$
(4.3)

Moreover, if the function $v\cdot q^{\prime}$ is constant on the interval
$[0,1]$, the asymptotic equivalence (4.3) also holds in the case $M=\infty$.

The proof of Theorem 4.5 is given in Appendix B.2. Note that in the Gaussian
white noise model we have $v\cdot q^{\prime}\equiv 1$, and in this case
Theorem 4.5 reproduces the result given in [BL96], at the top of page 2390.

###  Sketch of a different proof approach

The proofs of our results concerning Sobolev and Hölder classes in Sections
4.1 and 4.2 both relied on the general conditions stated in Theorem 3.1. We
will now sketch a different proof that is not based on the RKHS approach used
in this paper. It requires a slightly stronger assumption on the function $q$
and exploits results from the seminal paper [BL96], which are applied to
suitable transformations of the experiments of interest. Although checking all
the details of this alternative approach might be a tedious affair, we will
state the main steps here which yields a look on the asymptotic equivalence
from a different angle.

The arguments used in the sequel make use of several transformations of the
experiments. To be precise, starting with the discrete experiment
$\mathfrak{E}_{1,n}$, we note that observing the $Y_{i,n}$ in Equation (3.1)
is equivalent to observing

$Y^{\star}_{j,n}=\frac{\sum_{i=1}^{j}Y_{i,n}}{v(t_{j,n})}=\frac{\sum_{i=1}^{j}f(t_{i,n})}{v(t_{j,n})}+\sqrt{n}W_{q(t_{j,n})}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ j=1,\ldots,n.$

By taking increments these observations are in turn equivalent to the
experiment $\mathfrak{E}_{1,n}^{\star}$ with observations

$Y^{\star}_{j,n}-Y^{\star}_{j-1,n}=\mu_{j,n}+\sigma_{j,n}\varepsilon_{j,n}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ j=1,\ldots,n,$ (4.4)

where $Y^{\star}_{0,n}=0$, the $\varepsilon_{1,n},\ldots,\varepsilon_{n,n}$
are i.i.d. $\sim\mathcal{N}(0,1)$,

$\displaystyle\mu_{j,n}=\frac{\sum_{i=1}^{j}f(t_{i,n})}{v(t_{j,n})}-\frac{\sum_{i=1}^{j-1}f(t_{i,n})}{v(t_{j-1,n})},$

and

$\displaystyle\sigma_{j,n}^{2}=n\left[q(t_{j,n})-q(t_{j-1,n})\right].$

We now transform the continuous experiment $\mathfrak{E}_{2,n}$ to
observations with a similar structure as in (4.4) but different mean and
variance components. The observation of the path $Y_{t}$ in (3.2) is, assuming
that the function $v$ does not vanish, equivalent to observing the path

$Z_{t}^{\star}=\frac{Y(t)}{v(t)}=\int_{0}^{t}\mu(s)\mathrm{d}s+\frac{1}{\sqrt{n}}W_{q(t)},\qquad
t\in[0,1].$

Here, the function $\mu\colon[0,1]\to\mathbb{R}$ is given through

$\mu(s)=\frac{f(s)v(s)-v^{\prime}(s)\int_{0}^{s}f(\tau)\mathrm{d}\tau}{v^{2}(s)}.$

Using time-change for martingales (see [KS91], Chapter 3, Theorem 4.6) the
Gaussian process $Z_{t}^{\star}$ can be characterized by the SDE

$\mathrm{d}Z^{\star}(t)=\mu(t)\mathrm{d}(t)+\frac{1}{\sqrt{n}}\sqrt{q^{\prime}(t)}\mathrm{d}W_{t}.$
(4.5)

Now, under regularity assumptions on $\mu$ and $q$ Theorem 4.1 in [BL96] can
be applied to show asymptotic equivalence of the experiments where one either
observes the path $Z_{t}^{\star}$ or the discrete data

$Z_{j,n}^{\star}=\mu(j/(n+1))+\sqrt{q^{\prime}(j/(n+1))}\varepsilon_{j,n}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ j=1,\ldots,n,$ (4.6)

and $\varepsilon_{1,n},\ldots,\varepsilon_{n,n}$ are i.i.d.
$\sim\mathcal{N}(0,1)$. Let us denote the latter experiment by
$\mathfrak{E}_{2,n}^{\star}$. Then, by the arguments sketched above the target
equivalence $\mathfrak{E}_{1,n}\approx\mathfrak{E}_{2,n}$ follows from the
equivalence $\mathfrak{E}_{1,n}^{\star}\approx\mathfrak{E}_{2,n}^{\star}$. In
order to prove this statement, however, it is sufficient to show that

$n\cdot\sup_{j}\left(\mu(j/(n+1))-\mu_{j,n}\right)^{2}+n\sup_{j}\left(q^{\prime}(j/(n+1))-\sigma_{j,n}^{2}\right)^{2}\to
0,$

which can be established under mild regularity assumptions.

We emphasize that the assumptions for this alternative proof strategy and the
ones imposed by us in Theorems 4.1 and 4.5, respectively, do not coincide
entirely. In particular, Theorems 4.1 and 4.5 in the paper assume the
existence of a Hölder continuous first derivative $q^{\prime}$. On the other
hand Theorem 4.1 in [BL96], which is used to establish the asymptotic
equivalence of the models (4.5) and (4.6) in this section, requires the
existence of a second derivative of $q$ (see Condition (4.1) in this
reference). From this point of view, the RKHS approach uses (slightly) weaker
assumptions.

###  Non-equivalence for Brownian bridge

We conclude this section by showing that the asymptotic equivalence result
established in Theorem 4.1 cannot be transferred to arbitrary Gauss-Markov
processes. To be more precise, Theorem 4.6 below shows that for the special
case of the Brownian bridge the deficiency
$\delta(\mathfrak{E}_{1,n},\mathfrak{E}_{2,n})$ is bounded away from zero, and
as a consequence the two experiments are not asymptotically equivalent over
Sobolev spaces $\Theta(\beta,L)$ for arbitrary smoothness $\beta$ and $L>0$.
Recall that the covariance kernel of a Brownian Bridge $(B_{t})_{t\in[0,1]}$
satisfies the representation

$\operatorname{Cov}(B(s),B(t))=u(s)v(t),\qquad 0\leq s\leq t\leq 1,$

with $u(s)=s$, $v(t)=1-t$. Therefore $v(1)=0$, and in addition Conditions a
and b in Theorem 4.1 are not satisfied.

######  Theorem 4.6.

Consider the two sequences of experiments
$(\mathfrak{E}_{1,n})_{n\in\mathbb{N}}$ and
$(\mathfrak{E}_{2,n})_{n\in\mathbb{N}}$ defined in (3.1) and (3.2),
respectively, where the error process $(\Xi_{t})_{t\in[0,1]}$ is a Brownian
Bridge. Then, for an arbitrary smoothness index $\beta>0$, any $L>0$, and all
$n\in\mathbb{N}$,

$\delta(\mathfrak{E}_{1,n}(\Theta(\beta,L)),\mathfrak{E}_{2,n}(\Theta(\beta,L)))\geq
1/4,$

where $\Theta(\beta,L)$ is the Sobolev ellipsoid introduced in (4.1). In
particular, the experiments $(\mathfrak{E}_{1,n})_{n\in\mathbb{N}}$ and
$(\mathfrak{E}_{2,n})_{n\in\mathbb{N}}$ are _not_ asymptotically equivalent.

##  Appendix A Proof of Theorem 3.1

The proof consists in the consideration of two intermediate experiments, given
through Equations (A.1) and (A.2) below, that lie between $\mathfrak{E}_{1,n}$
and $\mathfrak{E}_{2,n}$.

First step: We first show that under Condition i in Theorem 3.1 the model
given by Equation (3.1) is asymptotically equivalent to observing

$\displaystyle
Y_{i,n}^{\prime}=n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s+\sqrt{n}\xi_{i,n},\quad
i=1,\ldots,n,$ (A.1)

where $\xi_{i,n}=\Xi_{t_{i},n}-\Xi_{t_{i-1,n}}$ are the increments of the
process $(\Xi_{t})_{t\in[0,1]}$. Under Assumption 2.1, we can take advantage
of the representation (2.3) and write

$\displaystyle\xi_{i,n}$
$\displaystyle=v(t_{i,n})W_{q(t_{i,n})}-v(t_{i-1,n})W_{q(t_{i-1,n})},$

and the experiment (3.1) can be written as

$Y_{i,n}=f(t_{i,n})+\sqrt{n}\left[v(t_{i,n})W_{q(t_{i,n})}-v(t_{i-1,n})W_{q(t_{i-1,n})}\right].$

Adding and subtracting $v(t_{i,n})W_{q(t_{i-1,n})}$, we get

$\displaystyle Y_{i,n}$
$\displaystyle=f(t_{i,n})+\sqrt{n}(v(t_{i,n})-v(t_{i-1,n}))W_{q(t_{i-1,n})}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\sqrt{n}v(t_{i,n})(W_{q(t_{i,n})}-W_{q(t_{i-1,n}))}.$

Similarly, (A.1) can be written as

$\displaystyle Y_{i,n}^{\prime}$
$\displaystyle=n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s+\sqrt{n}(v(t_{i,n})-v(t_{i-1,n}))W_{q(t_{i-1,n})}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\sqrt{n}v(t_{i,n})(W_{q(t_{i,n})}-W_{q(t_{i-1,n})}).$

For the sake of a transparent notation let
$\mathbf{P}^{\mathbf{Y}_{n}}=\mathbf{P}_{1,n}^{f}$ denote the distribution of
the vector $\mathbf{Y}_{n}=(Y_{1,n},\ldots,Y_{n,n})$, where we do not reflect
the dependence on $f$ in the notation. Similarly, let
$\mathbf{P}^{\mathbf{Y}_{n}^{\prime}}$ denote the distribution of the vector
$\mathbf{Y}_{n}^{\prime}=(Y^{\prime}_{1,n},\ldots,Y^{\prime}_{n,n})$. Note
that the squared total variation distance can be bounded by the Kullback-
Leibler divergence, and therefore we will derive a bound for the Kullback-
Leibler distance between the distributions $\mathbf{P}^{\mathbf{Y}_{n}}$ and
$\mathbf{P}^{\mathbf{Y}_{n}^{\prime}}$ by suitable conditioning. Denote with
$\mathscr{F}_{i,n}$ the $\sigma$-algebra generated by
$\left\\{W_{q({t})},t\leq t_{i,n}\right\\}$. We have

$\displaystyle\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{n}},\mathbf{P}^{\mathbf{Y}_{n}^{\prime}})$
$\displaystyle=\mathbf{E}[\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{n}},\mathbf{P}^{\mathbf{Y}_{n}^{\prime}}|\mathscr{F}_{n-1,n})]$
$\displaystyle=\mathbf{E}[\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{1:n-1,n}},\mathbf{P}^{\mathbf{Y}_{1:n-1,n}^{\prime}}|\mathscr{F}_{n-1,n})+\mathrm{KL}(\mathbf{P}^{Y_{n,n}},\mathbf{P}^{Y_{n,n}^{\prime}}|\mathscr{F}_{n-1,n})]$
$\displaystyle=\mathbf{E}[\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{1:n-1,n}},\mathbf{P}^{\mathbf{Y}_{1:n-1,n}^{\prime}}|\mathscr{F}_{n-1,n})]+\mathbf{E}[\mathrm{KL}(\mathbf{P}^{Y_{n,n}},\mathbf{P}^{Y_{n,n}^{\prime}}|\mathscr{F}_{n-1,n})]$
$\displaystyle=\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{1:n-1,n}},\mathbf{P}^{\mathbf{Y}_{1:n-1,n}^{\prime}})+\mathbf{E}[\mathrm{KL}(\mathbf{P}^{Y_{n,n}},\mathbf{P}^{Y_{n,n}^{\prime}}|\mathscr{F}_{n-1,n})],$

where we use the notation $\mathbf{Y}_{1:n-1,n}=(Y_{1,n},\ldots,Y_{n-1,n})$
and
$\mathbf{Y}_{1:n-1,n}^{\prime}=(Y^{\prime}_{1,n},\ldots,Y^{\prime}_{n-1,n})$.
Repeating the same argument, one obtains

$\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{n}},\mathbf{P}^{\mathbf{Y}_{n}^{\prime}})=\sum_{i=1}^{n}\mathbf{E}[\mathrm{KL}(\mathbf{P}^{Y_{i,n}},\mathbf{P}^{Y_{i,n}^{\prime}}|\mathscr{F}_{i-1,n})]].$

In order to study the terms
$\mathbf{E}[\mathrm{KL}(\mathbf{P}^{Y_{i,n}},\mathbf{P}^{Y_{i,n}^{\prime}}|\mathscr{F}_{i-1,n})]$,
note that

$\displaystyle
Y_{i,n}|\mathscr{F}_{i-1,n}\sim\mathcal{N}(\mu_{i},\sigma_{i}^{2})\qquad\text{and}\qquad
Y^{\prime}_{i,n}|\mathscr{F}_{i-1,n}\sim\mathcal{N}(\mu_{i}^{\prime},\sigma_{i}^{2}),$

where

$\displaystyle\mu_{i}=f(t_{i,n})+\sqrt{n}(v(t_{i,n})-v(t_{i-1,n}))W_{q(t_{i-1,n})},$
$\displaystyle\mu_{i}^{\prime}=n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s+\sqrt{n}(v(t_{i,n})-v(t_{i-1,n}))W_{q(t_{i-1,n})},\quad\text{and}$
$\displaystyle\sigma_{i}^{2}=nv^{2}(t_{i,n})(q(t_{i,n})-q(t_{i-1,n})).$

Here and in the following, $\mathcal{N}(\mu,\sigma^{2})$ denotes a normal
distribution with mean $\mu$ and variance $\sigma^{2}$. Using the fact that
the Kullback-Leibler divergence between two normal distributions with common
variance is given by

$\mathrm{KL}(\mathcal{N}(\mu_{1},\sigma^{2}),\mathcal{N}(\mu_{2},\sigma^{2}))=\frac{(\mu_{1}-\mu_{2})^{2}}{2\sigma^{2}},$

we have

$\mathrm{KL}(\mathbf{P}^{Y_{i,n}},\mathbf{P}^{Y_{i,n}^{\prime}}|\mathcal{F}_{i-1,n})=\frac{(f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s)^{2}}{2nv^{2}(t_{i,n})(q(t_{i,n})-q(t_{i-1,n}))}.$

This yields

$\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{n}},\mathbf{P}^{\mathbf{Y}_{n}^{\prime}})=\frac{1}{2n}\sum_{i=1}^{n}\frac{(f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s)^{2}}{v^{2}(t_{i,n})(q(t_{i,n})-q(t_{i-1,n}))},$

and
$\mathrm{KL}(\mathbf{P}^{\mathbf{Y}_{n}},\mathbf{P}^{\mathbf{Y}_{n}^{\prime}})\to
0$ holds if and only if Condition i holds. Consequently, the experiments (3.1)
and (A.1) are asymptotically equivalent in this case.

Second step: Let $I(t|\boldsymbol{y}_{n})$ denote the Kriging interpolator
which is defined as

$I(t|\boldsymbol{y}_{n})=\left(K_{\Xi}(t,t_{1,n}),K_{\Xi}\left(t,t_{2,n}\right),\ldots,K_{\Xi}(t,t_{n,n})\right)\operatorname{Cov}(\boldsymbol{\Xi}_{n})^{-1}\boldsymbol{y}_{n}^{\top}$
(A.2)

for $\boldsymbol{y}_{n}=(y_{1,n},\ldots,y_{n,n})$ and
$\boldsymbol{\Xi}_{n}=(\Xi_{t_{1,n}},\ldots,\Xi_{t_{n,n}})$ (the additional
condition that $v(1)\neq 0$ guarantees the invertibility of
$\boldsymbol{\Xi}_{n}$; see Lemma A.1 in [DPZ16] for an explicit formula for
the entries of the inverse matrix). By definition the Kriging predictor is
linear in the argument $\boldsymbol{y}_{n}$, and a simple argument shows the
interpolation property

$I(t_{j,n}|\boldsymbol{y}_{n})=y_{j,n}\leavevmode\nobreak\ \text{ for
}j=1,\ldots,n.$ (A.3)

The second step now consists in proving (exact, that is, non-asymptotic)
equivalence of the experiment defined by the discrete observations (A.1) and
the experiment defined by the continuous path

$\widetilde{Y}_{t}=I(t|\mathbf{F}_{f,n})+n^{-1/2}\Xi_{t},$ (A.4)

where $\mathbf{F}_{f,n}=(F_{f,n}(t_{1,n}),\ldots,F_{f,n}(t_{n,n}))$. Defining
the partial sums $S^{\prime}_{k,n}=\sum_{j=1}^{k}Y^{\prime}_{j,n}$ and
recalling the notation $\xi_{k,n}=\Xi_{t_{k,n}}-\Xi_{t_{k-1,n}}$ for the
increments of the the process $(\Xi_{t})_{t\in[0,1]}$, we have

$\displaystyle S^{\prime}_{k,n}=\sum_{j=1}^{k}Y^{\prime}_{j,n}$
$\displaystyle=n\int_{0}^{t_{k,n}}f(s)\mathrm{d}s+\sqrt{n}\sum_{j=1}^{k}\xi_{j,n}=nF_{f}\left(t_{k,n}\right)+\sqrt{n}\Xi_{t_{k,n}}$
(A.5)

where we used the interpolating property (A.3) and the the definition (A.4).
Let $(\Xi_{t}^{\prime})_{t\in[0,1]}$ be an independent copy of
$(\Xi_{t})_{t\in[0,1]}$, and set
$R_{t}=\Xi_{t}^{\prime}-I(t|\boldsymbol{\Xi}_{n}^{\prime})$ with
$\boldsymbol{\Xi}_{n}^{\prime}=(\Xi^{\prime}_{t_{1,n}},\ldots,\Xi^{\prime}_{t_{n,n}})$.
Then, the process

$I(t|\boldsymbol{\Xi}_{n})+R_{t},\quad t\in[0,1],$

follows the same law as $(\Xi_{t})_{t\in[0,1]}$ and
$(\Xi_{t}^{\prime})_{t\in[0,1]}$, which can can be checked by a comparison of
the covariance structure (indeed, this kind of construction is valid for any
centered Gaussian process). Then, observing the definition (A.4), we have

$\displaystyle\widetilde{Y}_{t}$
$\displaystyle=I(t|\mathbf{F}_{f,n})+n^{-1/2}\Xi_{t}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{L}}}{{=}}I(t|\mathbf{F}_{f,n})+n^{-1/2}I(t|\boldsymbol{\Xi}_{n})+n^{-1/2}R_{t}$
$\displaystyle=n^{-1}(I(t|\mathbf{S}^{\prime}_{n})+\sqrt{n}R_{t}),$

where we used the notation
$\mathbf{S}^{\prime}_{n}=(S^{\prime}_{1.n},\ldots,S^{\prime}_{n.n})$, equation
(A.5) and the linearity of the Kriging estimator. Therefore, the process
$(\widetilde{Y}_{t})_{t\in[0,1]}$ can be constructed from the vector
$\mathbf{Y}^{\prime}_{n}$. On the other hand, the observations
$Y^{\prime}_{1,n},\ldots,Y^{\prime}_{n,n}$ can be recovered from the
trajectory $(\widetilde{Y}_{t})_{t\in[0,1]}$ since for $t=t_{k,n}$ the
interpolation property (A.3) yields

$\displaystyle n\widetilde{Y}_{t_{k,n}}$
$\displaystyle=nF_{f,n}\left(t_{k,n}\right)+\sqrt{n}\Xi_{t_{k,n}}$
$\displaystyle=n\int_{0}^{t_{k,n}}f(s)\mathrm{d}s+\sqrt{n}(\Xi_{t_{k,n}}-\Xi_{0})$
$\displaystyle=n\int_{0}^{t_{k,n}}f(s)\mathrm{d}s+\sqrt{n}\sum_{j=1}^{k}\xi_{j,n},$

and one obtains $Y^{\prime}_{i,n}$ as
$Y^{\prime}_{i,n}=n\widetilde{Y}_{t_{i}}-n\widetilde{Y}_{t_{i-1}}$. Hence, the
process $(\widetilde{Y}_{t})_{t\in[0,1]}$ and the vector
$\mathbf{Y}^{\prime}_{n}$ contain the same information and the experiments
(A.1) and (A.4) are equivalent.

Third step: It remains to show that the experiment
$\widetilde{\mathfrak{E}}_{2,n}$ defined by the path in (A.4) is
asymptotically equivalent to the experiment $\mathfrak{E}_{2,n}$ defined by

$Y_{t}=\int_{0}^{t}f(s)\mathrm{d}s+n^{-1/2}\Xi_{t}.$

For this purpose we denote by $\mathbf{P}^{(Y_{t})_{t\in[0,1]}}$ and
$\mathbf{P}^{(\widetilde{Y}_{t})_{t\in[0,1]}}$ the distributions of the
processes $(Y_{t})_{t\in[0,1]}$ and $(\widetilde{Y}_{t})_{t\in[0,1]}$,
respectively, where the dependence on the parameter $f$ is again suppressed.
First, note that the representation of the Kriging estimator shows that the
function $t\to I(t|\mathbf{F}_{f,n})$ belongs to the RKHS $\mathcal{H}(\Xi)$
associated with the covariance kernel $K_{\Xi}$. Second, Condition ii in the
statement of Theorem 3.1 yields that the same holds true for the function
$F_{f,n}(\boldsymbol{\cdot})$. Using the fact that the squared total variation
distance can be bounded by the Kullback-Leibler divergence, we obtain

$\displaystyle\Delta^{2}(\mathfrak{E}_{2,n},\widetilde{\mathfrak{E}}_{2,n})$
$\displaystyle\leq\sup_{f\in\Theta}\mathrm{KL}(\mathbf{P}^{(Y_{t})_{t\in[0,1]}},\mathbf{P}^{(\widetilde{Y}_{t})_{t\in[0,1]}})$
$\displaystyle=\frac{n}{2}\sup_{f\in\Theta}\lVert
F_{f}(\boldsymbol{\cdot})-I(\boldsymbol{\cdot}\mid\mathbf{F}_{f,n})\rVert_{\mathcal{H}(\Xi)}^{2}$
$\displaystyle=\frac{n}{2}\sup_{f\in\Theta}\inf_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\left\lVert
F_{f}(\boldsymbol{\boldsymbol{\cdot}})-\sum_{j=1}^{n}\alpha_{j}K_{\Xi}\left(\boldsymbol{\cdot},t_{j,n}\right)\right\rVert_{\mathcal{H}(\Xi)}^{2}\to
0$

(the first equality follows from Lemma 2 in [SH14], the second from Theorem
13.1 in [Wen05]), which completes the proof of Theorem 3.1.

##  Appendix B Proofs of the results in Section 4

###  Proof of Theorem 4.1

The proof consists in checking the two conditions i and ii in Theorem 3.1.

Verification of condition i: We have to show that the expression

$\frac{1}{n}\sup_{f\in\Theta(\beta,L)}\sum_{i=1}^{n}\frac{(f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s)^{2}}{v^{2}(t_{i,n})(q(t_{i,n})-q(t_{i-1,n}))}$

converges to zero as $n\to\infty$. By the assumptions regarding the functions
$v$ and $q$ and an application of the mean value theorem this is equivalent to
the condition

$\sup_{f\in\Theta(\beta,L)}\sum_{i=1}^{n}\Big{(}f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s\Big{)}^{2}\to
0.$ (B.1)

Let $f\in\Theta(\beta,L)$ with Fourier expansion
$f(\boldsymbol{\cdot})=\sum_{k\in\mathbb{Z}}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot})$.
For any $K\in\mathbb{N}$ (the appropriate value of $K=K(n)$ for our purposes
will be specified below) we define the functions

$\displaystyle f_{K}(\boldsymbol{\cdot})$ $\displaystyle=\sum_{|k|\leq
K}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot})\text{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ and }$
$\displaystyle f_{K}^{\top}(\boldsymbol{\cdot})$
$\displaystyle=f(\boldsymbol{\cdot})-f_{K}(\boldsymbol{\cdot})=\sum_{|k|>K}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot}),$

respectively. Consequently,

$\displaystyle\sum_{i=1}^{n}\Big{(}f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s\Big{)}^{2}\leq
3\sum_{i=1}^{n}A_{i,n}^{2}+3\sum_{i=1}^{n}B_{i,n}^{2}+3\sum_{i=1}^{n}C_{i,n}^{2},$
(B.2)

where

$\displaystyle A_{i,n}$
$\displaystyle=f_{K}(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f_{K}(s)\mathrm{d}s,$
$\displaystyle B_{i,n}$ $\displaystyle=f_{K}^{\top}(t_{i,n}),\quad\text{and}$
$\displaystyle C_{i,n}$
$\displaystyle=n\int_{t_{i-1,n}}^{t_{i,n}}f_{K}^{\top}(s)\mathrm{d}s.$

We now show for $K=n$ that the estimates

$\displaystyle\sum_{i=1}^{n}A_{i,n}^{2}$
$\displaystyle=\mathcal{O}(\max\\{n^{-1},n^{1-2\beta}\\}),$ (B.3)
$\displaystyle\sum_{i=1}^{n}B_{i,n}^{2}$
$\displaystyle=\mathcal{O}(n^{1-2\beta}),$ (B.4)
$\displaystyle\sum_{i=1}^{n}C_{i,n}^{2}$
$\displaystyle=\mathcal{O}(n^{1-2\beta}),$ (B.5)

hold uniformly with respect to $f\in\Theta(\beta,L)$. Then, assertion (B.1)
follows from (B.2) and the assumption $\beta>1/2$.

Proof of (B.3): The quantity

$\frac{1}{n}\sum_{j=1}^{n}\lvert A_{j,n}\rvert^{2}$

can be interpreted as the (average) energy of the discrete signal
$A^{(n)}=(A_{1,n},\ldots,A_{n,n})$. Define

$F_{j}=\frac{1}{n}\sum_{k=1}^{n}A_{k,n}e^{-2\pi\mathrm{i}kj/n}$

as the discrete Fourier transform of the signal $A^{(n)}$, then Parseval’s
identity for the discrete Fourier transform yields

$\frac{1}{n}\sum_{j=1}^{n}\lvert A_{j,n}\rvert^{2}=\sum_{j=1}^{n}\lvert
F_{j}\rvert^{2},$

and we have to derive an estimate for $n\sum_{j=1}^{n}\lvert F_{j}\rvert^{2}$.
For this purpose, we recall the notation of $A_{j,n}$ and note that

$\displaystyle F_{j}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\Big{(}f_{K}(t_{k,n})-n\int_{t_{k-1}}^{t_{k}}f_{K}(s)\mathrm{d}s\Big{)}e^{-2\pi\mathrm{i}kj/n}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\left(\sum_{1\leq\lvert l\rvert\leq
K}\theta_{l}e^{-2\pi\mathrm{i}lk/n}-n\int_{t_{k-1}}^{t_{k}}\sum_{1\leq\lvert
l\rvert\leq
K}\theta_{l}e^{-2\pi\mathrm{i}ls}\mathrm{d}s\right)e^{-2\pi\mathrm{i}kj/n}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\left(\sum_{1\leq\lvert l\rvert\leq
K}\theta_{l}e^{-2\pi\mathrm{i}lk/n}+n\sum_{1\leq\lvert l\rvert\leq
K}\frac{\theta_{l}}{2\pi\mathrm{i}l}\left[e^{-2\pi\mathrm{i}lk/n}-e^{-2\pi\mathrm{i}l(k-1)/n}\right]\right)e^{-2\pi\mathrm{i}kj/n}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\left(\sum_{1\leq\lvert l\rvert\leq
K}\theta_{l}e^{-2\pi\mathrm{i}lk/n}+n\sum_{1\leq\lvert l\rvert\leq
K}\frac{\theta_{l}}{2\pi\mathrm{i}l}\left[1-e^{2\pi\mathrm{i}l/n}\right]e^{-2\pi\mathrm{i}lk/n}\right)e^{-2\pi\mathrm{i}kj/n}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\sum_{1\leq\lvert l\rvert\leq
K}\theta_{l}\left[1+\frac{n}{2\pi\mathrm{i}l}(1-e^{2\pi\mathrm{i}l/n})\right]e^{-2\pi\mathrm{i}kl/n}e^{-2\pi\mathrm{i}kj/n}.$

From now on, we take $K=n$ and write

$\displaystyle F_{j}^{+}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\sum_{l=1}^{n}\theta_{l}\left[1+\frac{n}{2\pi\mathrm{i}l}(1-e^{2\pi\mathrm{i}l/n})\right]e^{-2\pi\mathrm{i}kl/n}e^{-2\pi\mathrm{i}kj/n},\quad\text{and}$
$\displaystyle F_{j}^{-}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\sum_{l=-n}^{-1}\theta_{l}\left[1+\frac{n}{2\pi\mathrm{i}l}(1-e^{2\pi\mathrm{i}l/n})\right]e^{-2\pi\mathrm{i}kl/n}e^{-2\pi\mathrm{i}kj/n}.$

Since $\sum_{j=1}^{n}\lvert F_{j}\rvert^{2}\leq 2\sum_{j=1}^{n}\lvert
F_{j}^{+}\rvert^{2}+2\sum_{j=1}^{n}\lvert F_{j}^{-}\rvert^{2}$, it is
sufficient to consider $\sum_{j=1}^{n}\lvert F_{j}^{+}\rvert^{2}$ (the term
involving $F_{j}^{-}$ is treated analogously). We have

$\displaystyle F_{j}^{+}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\sum_{l=1}^{n}\theta_{l}\left[1+\frac{n}{2\pi\mathrm{i}l}(1-e^{2\pi\mathrm{i}l/n})\right]e^{-2\pi\mathrm{i}kl/n}e^{-2\pi\mathrm{i}kj/n}$
$\displaystyle=\frac{1}{n}\sum_{l=1}^{n}\theta_{l}\left[1+\frac{n}{2\pi\mathrm{i}l}(1-e^{2\pi\mathrm{i}l/n})\right]\sum_{k=1}^{n}e^{-2\pi\mathrm{i}kl/n}e^{-2\pi\mathrm{i}kj/n}$
$\displaystyle=\theta_{l(j)}\left[1+\frac{n}{2\pi\mathrm{i}l(j)}(1-e^{2\pi\mathrm{i}l(j)/n})\right]$

where $l(j)=n-j$ for $j=1,\ldots,n-1$ and $l(n)=n$. Here, we used the well-
known fact that for any integer $m\in\mathbb{Z}$

$\sum_{k=1}^{n}e^{-2\pi\mathrm{i}km/n}=\begin{cases}n,&\text{if }m\in
n\mathbb{Z},\\\ 0,&\text{if }m\notin n\mathbb{Z}.\end{cases}$ (B.6)

Thus, we obtain (uniformly with respect to $\Theta$)

$\displaystyle\sum_{j=1}^{n}\lvert F_{j}^{+}\rvert^{2}$
$\displaystyle=\sum_{j=1}^{n}\lvert\theta_{j}\rvert^{2}\left|1+\frac{n}{2\pi\mathrm{i}j}(1-e^{2\pi\mathrm{i}j/n})\right|^{2}$
$\displaystyle=\sum_{j=1}^{n}\lvert\theta_{j}\rvert^{2}\left|\frac{n}{2\pi\mathrm{i}j}(e^{2\pi\mathrm{i}j/n}-1-2\pi\mathrm{i}j/n)\right|^{2}$
$\displaystyle\asymp
n^{2}\sum_{j=1}^{n}\frac{\lvert\theta_{j}\rvert^{2}}{\lvert
j\rvert^{2}}\left|e^{2\pi\mathrm{i}j/n}-1-2\pi\mathrm{i}j/n\right|^{2}$
$\displaystyle\lesssim
n^{2}\sum_{j=1}^{n}\frac{\lvert\theta_{j}\rvert^{2}}{\lvert
j\rvert^{2}}\cdot\left|j/n\right|^{4}=n^{-2}\sum_{j=1}^{n}\lvert\theta_{j}\rvert^{2}\lvert
j\rvert^{2}$
$\displaystyle=n^{-2}\sum_{j=1}^{n}\lvert\theta_{j}\rvert^{2}\lvert
j\rvert^{2\beta}\lvert j\rvert^{2-2\beta}\leq
n^{-2}L^{2}\max\\{1,n^{2-2\beta}\\}$
$\displaystyle\lesssim\max\\{n^{-2},n^{-2\beta}\\}.$

An analogous argument for the term $\sum_{j=1}^{n}\lvert F_{j}^{-}\rvert^{2}$
proves the estimate (B.3).

Proof of (B.4): We have

$\displaystyle\sum_{j=1}^{n}\lvert f_{K}^{\top}(t_{j,n})\rvert^{2}$
$\displaystyle=\sum_{j=1}^{n}\left|\sum_{|k|>K}\theta_{k}\exp(-2\pi\mathrm{i}kj/n)\right|^{2}$
$\displaystyle\leq
2\sum_{j=1}^{n}\left|\sum_{k>K}\theta_{k}\exp(-2\pi\mathrm{i}kj/n)\right|^{2}+2\sum_{j=1}^{n}\left|\sum_{k<-K}\theta_{k}\exp(-2\pi\mathrm{i}kj/n)\right|^{2},$

and it is again sufficient to consider the sum running over $k>K$. Using (B.6)
again, we get

$\displaystyle\sum_{j=1}^{n}\left|\sum_{k>K}\theta_{k}\exp(-2\pi\mathrm{i}kj/n)\right|^{2}$
$\displaystyle=\sum_{k,l>K}\theta_{k}\overline{\theta_{l}}\sum_{j=1}^{n}\exp(-2\pi\mathrm{i}(k-l)j/n)$
$\displaystyle=n\sum_{\begin{subarray}{c}k,l>K\\\ k-l\in
n\mathbb{Z}\end{subarray}}\theta_{k}\overline{\theta_{l}}.$

Taking $K=n$ here as well yields

$\displaystyle\sum_{j=1}^{n}\left|\sum_{k>K}\theta_{k}\exp(-2\pi\mathrm{i}kj/n)\right|^{2}$
$\displaystyle=\sum_{m=n+1}^{2n}\left|\sum_{r=0}^{\infty}\theta_{m+rn}\right|^{2}.$

Now,

$\displaystyle\left|\sum_{r=0}^{\infty}\theta_{m+rn}\right|^{2}$
$\displaystyle\leq\left(\sum_{r=0}^{\infty}\lvert\theta_{m+rn}\rvert^{2}(m+rn)^{2\beta}\right)\left(\sum_{r=0}^{\infty}(m+rn)^{-2\beta}\right),$

and we obtain

$\displaystyle\sum_{m=n+1}^{2n}\left|\sum_{r=0}^{\infty}\theta_{m+rn}\right|^{2}$
$\displaystyle\leq nL^{2}\sum_{r=0}^{\infty}(n+rn)^{-2\beta}\lesssim
n^{1-2\beta}$

uniformly over $f\in\Theta(\beta,L)$, which establishes (B.4).

Proof of (B.5): Using Jensen’s inequality and Parseval’s identity we obtain

$\displaystyle\sum_{i=1}^{n}\left(n\int_{t_{i-1,n}}^{t_{i,n}}f_{K}^{\top}(s)\mathrm{d}s\right)^{2}$
$\displaystyle\leq
n\int_{0}^{1}(f_{K}^{\top}(s))^{2}\mathrm{d}s=n\sum_{|k|>K}\lvert\theta_{k}\rvert^{2}\leq
nL^{2}K^{-2\beta}$

uniformly over $f\in\Theta(\beta,L)$, which is of order $n^{1-2\beta}$ if we
choose $K=n$.

Verification of condition ii: We have to show that

$D_{n}:=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\lVert
F_{f}(\boldsymbol{\cdot})-\sum_{j=1}^{n}\alpha_{j}K_{\Xi}(\boldsymbol{\cdot},t_{j,n})\rVert_{\mathcal{H}(\Xi)}^{2}=o(n^{-1})$
(B.7)

uniformly over all $f\in\Theta(\beta,L)$. Via the isomorphism $\psi$
introduced in Equation (2.4) we have for $g=\psi F_{f}$

$\displaystyle D_{n}$
$\displaystyle=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\lVert
g(\boldsymbol{\cdot})-\sum_{j=1}^{n}\alpha_{j}v(t_{j,n})\mathbf{1}_{[0,q(t_{j,n})]}(\boldsymbol{\cdot})\rVert_{L^{2}([0,T])}^{2}$
$\displaystyle=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\lVert
g(\boldsymbol{\cdot})-\sum_{j=1}^{n}\alpha_{j}\mathbf{1}_{(q(t_{j-1,n}),q(t_{j,n})]}(\boldsymbol{\cdot})\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\int_{0}^{T}(g(u)-\sum_{j=1}^{n}\alpha_{j}\mathbf{1}_{(q(t_{j-1,n}),q(t_{j,n})]}(u))^{2}\mathrm{d}u$
$\displaystyle=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\int_{0}^{1}(g(q(w))-\sum_{j=1}^{n}\alpha_{j}\mathbf{1}_{(q(t_{j-1,n}),q(t_{j,n})]}(q(w)))^{2}q^{\prime}(w)\mathrm{d}w$
$\displaystyle=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\int_{0}^{1}(g(q(w))-\sum_{j=1}^{n}\alpha_{j}\mathbf{1}_{(t_{j-1,n},t_{j,n}]}(w))^{2}q^{\prime}(w)\mathrm{d}w.$

Assuming that $q^{\prime}$ is bounded from above we obtain

$\displaystyle D_{n}\leq C(q)$
$\displaystyle\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\int_{0}^{1}(g(q(w))-\sum_{j=1}^{n}\alpha_{j}\mathbf{1}_{(t_{j-1,n},t_{j,n}]}(w))^{2}\mathrm{d}w.$

Note that

$g(q(w))=g_{1}(q(w))-g_{2}(q(w))$

with

$\displaystyle g_{1}(q(w))=\frac{f(w)}{v(w)q^{\prime}(w)},\quad\text{and}\quad
g_{2}(q(w))=\frac{F_{f}(w)v^{\prime}(w)}{v^{2}(w)q^{\prime}(w)}.$

With $f_{n}(\boldsymbol{\cdot})=\sum_{|k|\leq
n}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot})$ (and
$f_{n}^{\top}(\boldsymbol{\cdot})=f(\boldsymbol{\cdot})-f_{n}(\boldsymbol{\cdot})$)
we define

$\displaystyle\alpha_{j}^{(1)}=\frac{f_{n}(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\quad\text{
and
}\quad\alpha_{j}^{(2)}=\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(t_{j,n})q^{\prime}(t_{j,n})}.$

Using these notations we get

$D_{n}=\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\lVert
F_{f}(\boldsymbol{\cdot})-\sum_{j=1}^{n}\alpha_{j}K_{\Xi}(\boldsymbol{\cdot},t_{j,n})\rVert_{\mathcal{H}(\Xi)}^{2}\lesssim
I_{1}+I_{2}$ (B.8)

where

$\displaystyle
I_{1}=\int_{0}^{1}\Big{(}g_{1}(q(w))-\sum_{j=1}^{n}\alpha_{j}^{(1)}\mathbf{1}_{[t_{j-1,n},t_{j,n})}(w)\Big{)}^{2}\mathrm{d}w,$
(B.9) $\displaystyle
I_{2}=\int_{0}^{1}\Big{(}g_{2}(q(w))-\sum_{j=1}^{n}\alpha_{j}^{(2)}\mathbf{1}_{[t_{j-1,n},t_{j,n})}(w)\Big{)}^{2}\mathrm{d}w.$
(B.10)

We investigate the two terms $I_{1}$ and $I_{2}$ separately.

_Bound for $I_{1}$_: We use the estimate

$\displaystyle I_{1}$ $\displaystyle\lesssim I_{11}+I_{12},$ (B.11)

where

$\displaystyle I_{11}$
$\displaystyle=\int_{0}^{1}\left(\frac{f_{n}(w)}{v(w)q^{\prime}(w)}-\sum_{j=1}^{n}\alpha_{j}^{(1)}\mathbf{1}_{[t_{j-1,n},t_{j,n})}(w)\right)^{2}\mathrm{d}w,$
$\displaystyle I_{12}$
$\displaystyle=\int_{0}^{1}\left(\frac{f_{n}^{\top}(w)}{v(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w.$

For the first integral $I_{11}$ on the right-hand side of (B.11), we have

$\displaystyle I_{11}$
$\displaystyle=\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f_{n}(w)}{v(w)q^{\prime}(w)}-\frac{f_{n}(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w$
$\displaystyle\lesssim\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f_{n}(w)}{v(w)q^{\prime}(w)}-\frac{f_{n}(t_{j,n})}{v(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w$
(B.12)
$\displaystyle+\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f_{n}(t_{j,n})}{v(w)q^{\prime}(w)}-\frac{f_{n}(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w.$
(B.13)

First, we further decompose (B.12) as

$\displaystyle\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f_{n}(w)}{v(w)q^{\prime}(w)}-\frac{f_{n}(t_{j,n})}{v(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w$
$\displaystyle\lesssim
C(v,q)\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}(f_{n}(w)-f_{n}(t_{j,n}))^{2}\mathrm{d}w$
$\displaystyle\lesssim\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\lvert
f_{n}^{+}(w)-f_{n}^{+}(t_{j,n})\rvert^{2}\mathrm{d}w+\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\lvert
f_{n}^{-}(w)-f_{n}^{-}(t_{j,n})\rvert^{2}\mathrm{d}w\leavevmode\nobreak\ ,$

where

$\displaystyle
f_{n}^{+}=\sum_{k=1}^{n}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot})\quad\text{and}\quad
f_{n}^{-}=\sum_{k=-n}^{-1}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(\boldsymbol{\cdot}).$

In the sequel, we consider only the term involving $f_{n}^{+}$ since the sum
involving $f_{n}^{-}$ can be bounded using the same argument. We have the
identity

$\displaystyle\lvert f_{n}^{+}(w)-f_{n}^{+}(t_{j,n})\rvert^{2}$
$\displaystyle=\lvert\sum_{k=1}^{n}\theta_{k}(\boldsymbol{\mathrm{e}}_{k}(w)-\boldsymbol{\mathrm{e}}_{k}(t_{j,n}))\rvert^{2}$
$\displaystyle=\sum_{k,l=1}^{n}\theta_{k}\overline{\theta}_{l}\exp(-2\pi\mathrm{i}kj/n)[\exp(2\pi\mathrm{i}k(j/n-w))-1]$
$\displaystyle\hskip
50.00008pt\cdot\exp(2\pi\mathrm{i}lj/n)[\exp(-2\pi\mathrm{i}l(j/n-w))-1].$

From this identity we obtain (exploiting (B.6) again)

$\displaystyle\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\lvert
f_{n}^{+}(w)-f_{n}^{+}(t_{j,n})\rvert^{2}\mathrm{d}w$ $\displaystyle\hskip
10.00002pt=\sum_{j=1}^{n}\sum_{k,l=1}\theta_{k}\overline{\theta}_{l}\exp(2\pi\mathrm{i}(l-k)j/n)\int_{\frac{j-1}{n}}^{\frac{j}{n}}[\exp(2\pi\mathrm{i}k(j/n-w))-1]$
$\displaystyle\hskip
213.39566pt\cdot[\exp(-2\pi\mathrm{i}l(j/n-w))-1]\mathrm{d}w$
$\displaystyle\hskip
10.00002pt=\sum_{j=1}^{n}\sum_{k,l=1}\theta_{k}\overline{\theta}_{l}\exp(2\pi\mathrm{i}(l-k)j/n)\int_{0}^{\frac{1}{n}}[\exp(2\pi\mathrm{i}k(1/n-w))-1]$
$\displaystyle\hskip
214.81839pt\cdot[\exp(-2\pi\mathrm{i}l(1/n-w))-1]\mathrm{d}w$
$\displaystyle\hskip
10.00002pt=n\sum_{k=1}^{n}|\theta_{k}|^{2}\int_{0}^{\frac{1}{n}}[\exp(2\pi\mathrm{i}k(1/n-w))-1][\exp(-2\pi\mathrm{i}k(1/n-w))-1]\mathrm{d}w$
$\displaystyle\hskip 10.00002pt\leq
Cn\sum_{k=1}^{n}|\theta_{k}|^{2}k^{2}n^{-3}$ $\displaystyle\hskip
10.00002pt\leq Cn^{-2}\sum_{k=1}^{n}|\theta_{k}|^{2}k^{2\beta}k^{-2\beta+2}$
$\displaystyle\hskip 10.00002pt\leq
C(L)n^{-2}\max\\{1,n^{-2\beta+2}\\}\lesssim\max\\{n^{-2},n^{-2\beta}\\}=o(n^{-1}).$

To derive an estimate of (B.13) we note that for any $n\in\mathbb{N}$,

$\displaystyle\lVert
f_{n}\rVert^{2}_{\infty}=\sup_{x\in[0,1]}\lvert\sum_{|k|\leq
n}\theta_{k}\boldsymbol{\mathrm{e}}_{k}(x)\rvert^{2}$
$\displaystyle\leq\left(\sum_{|k|\leq
n}\lvert\theta_{k}\rvert^{2}k^{2\beta}\right)\left(\sum_{|k|\leq
n}k^{-2\beta}\right)\leq C(L,\beta)$

(the same estimate holding true for $f$ instead of $f_{n}$ which formally
corresponds to $n=\infty$). Hence,

$\displaystyle\sum_{j=1}^{n}$
$\displaystyle\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f_{n}(t_{j,n})}{v(w)q^{\prime}(w)}-\frac{f_{n}(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w$
$\displaystyle=\sum_{j=1}^{n}\lvert
f_{n}(t_{j,n})\rvert^{2}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{1}{v(w)q^{\prime}(w)}-\frac{1}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w$
$\displaystyle\leq
C(L,\beta)\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left|\frac{v(t_{j,n})q^{\prime}(t_{j,n})-v(w)q^{\prime}(w)}{v(w)q^{\prime}(w)v(t_{j,n})q^{\prime}(t_{j,n})}\right|^{2}\mathrm{d}w$
$\displaystyle\leq
C(L,\beta,v,q)\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\lvert
v(t_{j,n})q^{\prime}(t_{j,n})-v(w)q^{\prime}(w)\rvert^{2}\mathrm{d}w.$

Because the product of a $\gamma_{1}$-Hölder function and a
$\gamma_{2}$-Hölder function is (at least) Hölder with index
$\min\\{\gamma_{1},\gamma_{2}\\}$ we obtain from our assumptions that the
function $vq^{\prime}$ is Hölder continuous with index $\gamma>1/2$. Thus,

$\displaystyle\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f_{n}(t_{j,n})}{v(w)q^{\prime}(w)}-\frac{f_{n}(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w$
$\displaystyle\leq
C(L,\beta,v,q)\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}n^{-2\gamma}\mathrm{d}w$
$\displaystyle=C(L,\beta,v,q)n^{-2\gamma},$

and this is $o(n^{-1})$ since $\gamma>1/2$. Combining these arguments we
obtain

$I_{11}=o(n^{-1}).$

Finally, the second integral $I_{12}$ on the right-hand side of (B.11) can
bounded as follows:

$\displaystyle I_{12}$
$\displaystyle=\int_{0}^{1}\left(\frac{f_{n}^{\top}(w)}{v(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w\leq
C(v,q)\int_{0}^{1}\lvert f_{n}^{\top}(w)\rvert^{2}\mathrm{d}w$
$\displaystyle\leq C(v,q)\sum_{\lvert k\rvert>n}\lvert\theta_{k}\rvert^{2}\leq
C(v,q)L^{2}n^{-2\beta}=o(n^{-1}).$

Observing the estimate (B.1) we finally obtain $I_{1}=o(n^{-1})$.

_Bound for $I_{2}$_: In analogy to the decomposition of the term $I_{11}$ on
the right-hand side of (B.11) we have

$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{1}\left(\frac{F_{f}(w)v^{\prime}(w)}{v^{2}(w)q^{\prime}(w)}-\sum_{j=1}^{n}\alpha_{j}^{(2)}\mathbf{1}_{[t_{j-1,n},t_{j,n})}(w)\right)^{2}\mathrm{d}w$
$\displaystyle\lesssim\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{F_{f}(w)v^{\prime}(w)}{v^{2}(w)q^{\prime}(w)}-\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w$
(B.14) $\displaystyle\hskip
10.00002pt+\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(w)q^{\prime}(w)}-\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w.$
(B.15)

Note that $F_{f}$ is Lipschitz since it is continuously differentiable (recall
that $f$ itself is continuous since $\beta>1/2$) and $v^{\prime}$ is Hölder
with index $\gamma>1/2$ due to our assumptions. Thus, the term (B.14) can be
bounded as

$\displaystyle\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{F_{f}(w)v^{\prime}(w)}{v^{2}(w)q^{\prime}(w)}-\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w\leq
C(v,q)n^{-2\gamma}$

which is of order $o(n^{-1})$. For the term (B.15) we obtain using our
assumptions that

$\displaystyle\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(w)q^{\prime}(w)}-\frac{F_{f}(t_{j,n})v^{\prime}(t_{j,n})}{v^{2}(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w$
$\displaystyle\leq
C(\beta,L,v,q)\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}(v^{2}(t_{j,n})q^{\prime}(t_{j,n})-v^{2}(w)q^{\prime}(w))^{2}\mathrm{d}w.$

Using the same arguments as for the bound of (B.13), this term can be shown to
be of order $o(n^{-1})$. Since both terms $I_{1}$ and $I_{2}$ are of order
$o(n^{-1})$ the assertion (B.7) follows from (B.8).

###  Proof of Theorem 4.5

As in the proof of Theorem 4.1, we have to verify the the two conditions i and
ii from Theorem 3.1.

Verification of condition i: As in the Sobolev case it is sufficient to show
that

$\sup_{f\in\mathcal{F}(\alpha,L,M)}\sum_{i=1}^{n}\Big{(}f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s\Big{)}^{2}\to
0.$

By the mean value theorem
$n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s=f(\zeta_{i,n})$ for some
$t_{i-1,n}\leq\zeta_{i,n}\leq t_{i,n}$. Thus, since
$f\in\mathcal{F}(\alpha,L,M)$,

$\displaystyle\sum_{i=1}^{n}\Big{(}f(t_{i,n})-n\int_{t_{i-1,n}}^{t_{i,n}}f(s)\mathrm{d}s\Big{)}^{2}$
$\displaystyle=\sum_{i=1}^{n}(f(t_{i,n})-f(\zeta_{i,n}))^{2}$
$\displaystyle\leq\sum_{i=1}^{n}L^{2}\lvert t_{i,n}-\zeta_{i,n}\rvert^{2}$
$\displaystyle\leq L^{2}n^{-2\alpha+1},$

and the last term converges to zero uniformly over
$f\in\mathcal{F}(\alpha,L,M)$ whenever $\alpha>1/2$.

Verification of condition ii: The proof is based on nearly the same reduction
as in the Sobolev case. Again, we consider the bound

$\min_{{\boldsymbol{\alpha}}_{n}\in\mathbb{R}^{n}}\lVert
F_{f}(\boldsymbol{\cdot})-\sum_{j=1}^{n}\alpha_{j}K_{\Xi}(\boldsymbol{\cdot},t_{j,n})\rVert_{\mathcal{H}(\Xi)}^{2}\lesssim
I_{1}+I_{2}$ (B.16)

where we define $I_{1}$ and $I_{2}$ as in Appendix B.1 (see the equations
(B.9) and (B.10)) with the only exception that we now put

$\alpha_{j}^{(1)}=\frac{f(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}$

in the definition of $I_{1}$. Then,

$\displaystyle I_{1}$
$\displaystyle\lesssim\int_{0}^{1}\left(\frac{f(w)}{v(w)q^{\prime}(w)}-\sum_{j=1}^{n}\alpha_{j}^{(1)}\mathbf{1}_{[t_{j-1,n},t_{j,n})}(w)\right)^{2}\mathrm{d}w$
$\displaystyle=\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f(w)}{v(w)q^{\prime}(w)}-\frac{f(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w$
$\displaystyle\lesssim\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f(w)}{v(w)q^{\prime}(w)}-\frac{f(t_{j,n})}{v(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w$
$\displaystyle\hskip
10.00002pt+\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f(t_{j,n})}{v(w)q^{\prime}(w)}-\frac{f(t_{j,n})}{v(t_{j,n})q^{\prime}(t_{j,n})}\right)^{2}\mathrm{d}w.$

The first integral can be bounded as

$\displaystyle\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}\left(\frac{f(w)}{v(w)q^{\prime}(w)}-\frac{f(t_{j,n})}{v(w)q^{\prime}(w)}\right)^{2}\mathrm{d}w$
$\displaystyle\leq
C(v,q)\sum_{j=1}^{n}\int_{t_{j-1,n}}^{t_{j,n}}(f(w)-f(t_{j,n}))^{2}\mathrm{d}w$
$\displaystyle\leq C(v,q)L^{2}n^{-2\alpha},$

which converges to zero as $n$ increases. The second integral can be bounded
as in the Sobolev case using the assumption that $f$ is bounded (as one can
easily see, the assumption of uniform boundedness can be dropped if $v\cdot
q^{\prime}$ is constant; this is for instance satisfied in the case of
Brownian motion). Hence, $I_{1}$ converges to $0$.

The term $I_{2}$ in (B.16) can be bounded exactly as the corresponding term in
the Sobolev ellipsoid case. This finishes the proof of the theorem.

###  Proof of Theorem 4.6

To prepare the proof, we recall an alternative (but equivalent)
characterization of asymptotic equivalence in the framework of statistical
decision theory. Let
$\mathfrak{E}=(\mathcal{X},\mathscr{X},(\mathbf{P}_{\theta})_{\theta\in\Theta})$
be a statistical experiment. In decision theory one considers a decision space
$(\mathcal{A},\mathscr{A})$ where the set $\mathcal{A}$ contains the potential
decisions (or actions) that are at the observers disposal and $\mathscr{A}$ is
a $\sigma$-field on $\mathcal{A}$. In addition, there is a _loss function_

$\ell\colon\Theta\times\mathcal{A}\to[0,\infty),\qquad(\theta,a)\mapsto\ell(\theta,a)$

with the interpretation that a loss $\ell(\theta,a)$ occurs if the
statistician chooses the action $a\in{\cal A}$ and $\theta\in\Theta$ is the
true state of nature. A (randomized) decision rule is a Markov kernel
$\rho\colon\mathcal{X}\times\mathscr{A}\to[0,1]$, and the associated _risk_ is

$R_{\theta}(\mathfrak{E},\rho,\ell)=\int_{\mathcal{X}}\left(\int_{\mathcal{A}}\ell(\theta,a)\rho(x,\mathrm{d}a)\right)\mathbf{P}_{\theta}(\mathrm{d}x).$

Then, the deficiency between two experiments $\mathfrak{E}_{1}$ and
$\mathfrak{E}_{2}$ is exactly the quantity

$\delta(\mathfrak{E}_{1},\mathfrak{E}_{2})=\inf_{\rho_{1}}\sup_{\rho_{2}}\sup_{\theta}\sup_{\ell}\lvert
R_{\theta}(\mathfrak{E}_{1},\rho_{1},\ell)-R_{\theta}(\mathfrak{E}_{2},\rho_{2},\ell)\rvert$
(B.17)

(see [Mar16], Theorem 2.7, and the references cited there), where the supremum
is taken over all loss functions $\ell$ with $0\leq\ell(\theta,a)\leq 1$ for
all $\theta\in\Theta$ and $a\in\mathcal{A}$, all admissible parameters
$\theta\in\Theta$, and decision rules $\rho_{2}$ in the second experiment. The
infimum is taken over all decision rules $\rho_{1}$ in the first experiment.

After these preliminaries, let us go on to the proof of the theorem. We
consider the decision space
$(\mathcal{A},\mathscr{A})=(\mathbb{R},\mathscr{B}(\mathbb{R}))$ and the loss
function

$\ell\colon\Theta(\beta,L)\times\mathcal{A}\to\\{0,1\\},\quad(f,a)\mapsto\ell(f,a)=\begin{cases}1,&\text{if
}\int_{0}^{1}f(x)\mathrm{d}x\neq a,\\\ 0,&\text{if
}\int_{0}^{1}f(x)\mathrm{d}x=a.\end{cases}$

In the experiment $\mathfrak{E}_{2,n}$ we observe the whole path
$Y=\\{Y_{t},\,t\in[0,1]\\}$ satisfying

$Y_{t}=\int_{0}^{t}f(s)\mathrm{d}s+\frac{1}{\sqrt{n}}B_{t},\quad
t\in[0,1]\leavevmode\nobreak\ ,$

where $(B_{t})_{t\in[0,1]}$ is a Brownian Bridge, and we consider the (non-
randomized) decision rule $\rho_{2}$ defined by

$\rho_{2}(h)=h(1)-h(0)\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ~h\in{\cal
C}([0,1],\mathbb{R}).$

This directly yields $\rho_{2}(Y)=\int_{0}^{1}f(s)\mathrm{d}s$ since
$\Xi_{0}=\Xi_{1}=0$ for the Brownian bridge. Hence,

$R_{f}(\mathfrak{E}_{2,n},\rho_{2},\ell)=\int\ell(f,\rho_{2}(Y))\mathbf{P}^{f}_{2,n}(\mathrm{d}Y)=0$

for all $f\in\Theta(\beta,L)$ and $n\in\mathbb{N}$. From (B.17) we thus obtain

$\displaystyle\delta(\mathfrak{E}_{1,n},\mathfrak{E}_{2,n})$
$\displaystyle\geq\inf_{\rho_{1}}\sup_{f\in\Theta(\beta,L)}\lvert
R_{f}(\mathfrak{E}_{1,n},\rho_{1},\ell)\rvert.$

Recall the notation $\boldsymbol{\mathrm{e}}_{n}(x)=\exp(-2\pi\mathrm{i}nx)$
and introduce the functions $f_{0}\equiv 0$ and

$f_{n}(x)=\sqrt{\frac{2}{3}}\frac{L}{n^{\beta}}\left[1-\frac{1}{2}\boldsymbol{\mathrm{e}}_{n}(x)-\frac{1}{2}\boldsymbol{\mathrm{e}}_{-n}(x)\right]=\sqrt{\frac{2}{3}}\frac{L}{n^{\beta}}\left[1-\frac{1}{2}\cos(2\pi
nx)\right]$

for $n\in\mathbb{N}$. It is easily seen that $f_{n}$ belongs to
$\Theta(\beta,L)$. Note that by construction $f_{n}(j/n)=0$ for $j=1,\ldots,n$
and thus in the experiment $\mathfrak{E}_{1,n}$ the identity
$\mathbf{P}_{1,n}^{f_{0}}=\mathbf{P}_{1,n}^{f_{n}}$ holds. For the considered
loss function we have

$\displaystyle R_{f}(\mathfrak{E}_{1,n},\rho_{1},\ell)$
$\displaystyle=\int_{\mathbb{R}^{n}}\rho_{1}(\mathbf{Y}_{n})\left(\mathbb{R}\backslash\left\\{\int_{0}^{1}f(x)\mathrm{d}x\right\\}\right)\mathbf{P}_{1,n}^{f}(\mathrm{d}\mathbf{Y}_{n}),$

where $\rho_{1}$ is any (potentially randomized) decision rule. Because

$\int_{0}^{1}f_{0}(x)\mathrm{d}x=0\neq\sqrt{2/3}Ln^{-\beta}=\int_{0}^{1}f_{n}(x)\mathrm{d}x$

at least one of the quantities
$\rho_{1}(\mathbf{Y}_{n})(\mathbb{R}\backslash\\{\int_{0}^{1}f_{0}(x)\mathrm{d}x\\})$
and
$\rho_{1}(\mathbf{Y}_{n})(\mathbb{R}\backslash\\{\int_{0}^{1}f_{n}(x)\mathrm{d}x\\})$
must be $\geq 1/2$ for any $\mathbf{Y}_{n}$ (otherwise there is a
contradiction). Thus, setting
$A_{\diamond}=\\{\mathbf{Y}_{n}\in\mathbb{R}^{n}:\rho_{1}(\mathbf{Y}_{n})(\mathbb{R}\backslash\\{\int_{0}^{1}f_{\diamond}(x)\mathrm{d}x\\})\geq
1/2\\}$ for $\diamond\in\\{0,n\\}$ one has $A_{0}\cup A_{n}=\mathbb{R}^{n}$.
As a consequence, either
$\mathbf{P}_{1,n}^{f_{0}}(A_{0})=\mathbf{P}_{1,n}^{f_{n}}(A_{0})\geq 1/2$ or
$\mathbf{P}_{1,n}^{f_{0}}(A_{n})=\mathbf{P}_{1,n}^{f_{n}}(A_{n})\geq 1/2$
holds. Without loss of generality, we assume that
$\mathbf{P}_{1,n}^{f_{0}}(A_{0})=\mathbf{P}_{1,n}^{f_{n}}(A_{0})\geq 1/2$ (the
other case follows by exactly the same argument). In this case, using the
definition of the set $A_{0}$,

$\displaystyle\delta(\mathfrak{E}_{1,n},\mathfrak{E}_{2,n})$
$\displaystyle\geq\inf_{\rho_{1}}\sup_{f\in\\{f_{0},f_{n}\\}}R_{f}(\mathfrak{E}_{1,n},\rho_{1},\ell)$
$\displaystyle\geq\inf_{\rho_{1}}\int_{\mathbb{R}^{n}}\rho_{1}(\mathbf{Y}_{n})\left(\mathbb{R}\backslash\left\\{\int_{0}^{1}f_{0}(x)\mathrm{d}x\right\\}\right)\mathbf{P}_{1,n}^{f_{0}}(\mathrm{d}\mathbf{Y}_{n})$
$\displaystyle\geq\inf_{\rho_{1}}\int_{A_{0}}\rho_{1}(\mathbf{Y}_{n})\left(\mathbb{R}\backslash\left\\{\int_{0}^{1}f_{0}(x)\mathrm{d}x\right\\}\right)\mathbf{P}_{1,n}^{f_{0}}(\mathrm{d}\mathbf{Y}_{n})$
$\displaystyle\geq\frac{1}{2}\mathbf{P}_{1,n}^{f_{0}}(A_{0})$
$\displaystyle\geq\frac{1}{4},$

which proves the assertion.

## References

  * [Bed87] Jay H. Beder  “A sieve estimator for the mean of a Gaussian process”  In _Ann. Statist._ 15.1, 1987, pp. 59–78  DOI: 10.1214/aos/1176350253
  * [BL96] Lawrence D. Brown and Mark G. Low  “Asymptotic equivalence of nonparametric regression and white noise”  In _Ann. Statist._ 24.6, 1996, pp. 2384–2398  DOI: 10.1214/aos/1032181159
  * [Bro+02] Lawrence D. Brown, T. Tony Cai, Mark G. Low and Cun-Hui Zhang  “Asymptotic equivalence theory for nonparametric regression with random design” Dedicated to the memory of Lucien Le Cam  In _Ann. Statist._ 30.3, 2002, pp. 688–707  DOI: 10.1214/aos/1028674838
  * [BTA04] Alain Berlinet and Christine Thomas-Agnan  “Reproducing kernel Hilbert spaces in probability and statistics” With a preface by Persi Diaconis  Kluwer Academic Publishers, Boston, MA, 2004, pp. xxii+355  DOI: 10.1007/978-1-4419-9096-9
  * [BZ98] Lawrence D. Brown and Cun-Hui Zhang  “Asymptotic nonequivalence of nonparametric experiments when the smoothness index is $1/2$”  In _Ann. Statist._ 26.1, 1998, pp. 279–287  DOI: 10.1214/aos/1030563986
  * [Car06] Andrew V. Carter  “A continuous Gaussian approximation to a nonparametric regression in two dimensions”  In _Bernoulli_ 12.1, 2006, pp. 143–156
  * [Car07] Andrew V. Carter  “Asymptotic approximation of nonparametric regression experiments with unknown variances”  In _Ann. Statist._ 35.4, 2007, pp. 1644–1673  DOI: 10.1214/009053606000001613
  * [Car09] Andrew V. Carter  “Asymptotically sufficient statistics in nonparametric regression experiments with correlated noise”  In _J. Probab. Stat._ , 2009, pp. Art. ID 275308, 19  DOI: 10.1155/2009/275308
  * [Doo49] J. L. Doob  “Heuristic approach to the Kolmogorov-Smirnov theorems”  In _Ann. Math. Statistics_ 20, 1949, pp. 393–403  DOI: 10.1214/aoms/1177729991
  * [DPZ16] Holger Dette, Andrey Pepelyshev and Anatoly Zhigljavsky  “Optimal designs in regression with correlated errors”  In _Ann. Statist._ 44.1, 2016, pp. 113–152  DOI: 10.1214/15-AOS1361
  * [GN06] Ion G. Grama and Michael H. Neumann  “Asymptotic equivalence of nonparametric autoregression and nonparametric regression”  In _Ann. Statist._ 34.4, 2006, pp. 1701–1732  DOI: 10.1214/009053606000000560
  * [GN98] Ion Grama and Michael Nussbaum  “Asymptotic equivalence for nonparametric generalized linear models”  In _Probab. Theory Related Fields_ 111.2, 1998, pp. 167–214  DOI: 10.1007/s004400050166
  * [GNZ10] Georgi K. Golubev, Michael Nussbaum and Harrison H. Zhou  “Asymptotic equivalence of spectral density estimation and Gaussian white noise”  In _Ann. Statist._ 38.1, 2010, pp. 181–214  DOI: 10.1214/09-AOS705
  * [Joh99] Iain M. Johnstone  “Wavelet shrinkage for correlated data and inverse problems: adaptivity results”  In _Statist. Sinica_ 9.1, 1999, pp. 51–83
  * [JS97] Iain M. Johnstone and Bernard W. Silverman  “Wavelet threshold estimators for data with correlated noise”  In _J. Roy. Statist. Soc. Ser. B_ 59.2, 1997, pp. 319–351  DOI: 10.1111/1467-9868.00071
  * [KS91] Ioannis Karatzas and Steven E. Shreve  “Brownian motion and stochastic calculus” 113, Graduate Texts in Mathematics  Springer-Verlag, New York, 1991, pp. xxiv+470  DOI: 10.1007/978-1-4612-0949-2
  * [LC86] Lucien Le Cam  “Asymptotic methods in statistical decision theory”, Springer Series in Statistics  Springer-Verlag, New York, 1986, pp. xxvi+742  DOI: 10.1007/978-1-4612-4946-7
  * [LCY00] Lucien Le Cam and Grace Lo Yang  “Asymptotics in statistics” Some basic concepts, Springer Series in Statistics  Springer-Verlag, New York, 2000, pp. xiv+285  DOI: 10.1007/978-1-4612-1166-2
  * [Mar16] Ester Mariucci  “Le Cam theory on the comparison of statistical models”  In _Grad. J. Math._ 1.2, 2016, pp. 81–91
  * [Mei11] Alexander Meister  “Asymptotic equivalence of functional linear regression and a white noise inverse problem”  In _Ann. Statist._ 39.3, 2011, pp. 1471–1495  DOI: 10.1214/10-AOS872
  * [MM65] C. B. Mehr and J. A. McFadden  “Certain properties of Gaussian processes and their first-passage times”  In _J. Roy. Statist. Soc. Ser. B_ 27, 1965, pp. 505–522  URL: http://links.jstor.org/sici?sici=0035-9246(1965)27:3<505:CPOGPA>2.0.CO;2-8&origin=MSN
  * [Nev68] Jacques Neveu  “Processus aléatoires gaussiens”, Séminaire de Mathématiques Supérieures, No. 34 (Été, 1968)  Les Presses de l’Université de Montréal, Montreal, Que., 1968, pp. 224
  * [Nus96] Michael Nussbaum  “Asymptotic equivalence of density estimation and Gaussian white noise”  In _Ann. Statist._ 24.6, 1996, pp. 2399–2430  DOI: 10.1214/aos/1032181160
  * [PR16] Vern I. Paulsen and Mrinal Raghupathi  “An introduction to the theory of reproducing kernel Hilbert spaces” 152, Cambridge Studies in Advanced Mathematics  Cambridge University Press, Cambridge, 2016, pp. x+182  DOI: 10.1017/CBO9781316219232
  * [Rei08] Markus Reiß  “Asymptotic equivalence for nonparametric regression with multivariate and random design”  In _Ann. Statist._ 36.4, 2008, pp. 1957–1982  DOI: 10.1214/07-AOS525
  * [Rei11] Markus Reiß  “Asymptotic equivalence for inference on the volatility from noisy observations”  In _Ann. Statist._ 39.2, 2011, pp. 772–802  DOI: 10.1214/10-AOS855
  * [Roh04] Angelika Rohde  “On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise”  In _Statist. Decisions_ 22.3, 2004, pp. 235–243  DOI: 10.1524/stnd.22.3.235.57063
  * [SH14] Johannes Schmidt-Hieber  “Asymptotic equivalence for regression under fractional noise”  In _Ann. Statist._ 42.6, 2014, pp. 2557–2585  DOI: 10.1214/14-AOS1262
  * [Sle61] D. Slepian  “First passage time for a particular Gaussian process”  In _Ann. Math. Statist._ 32, 1961, pp. 610–612  DOI: 10.1214/aoms/1177705068
  * [Tor91] Erik Torgersen  “Comparison of statistical experiments” 36, Encyclopedia of Mathematics and its Applications  Cambridge University Press, Cambridge, 1991, pp. xx+675  DOI: 10.1017/CBO9780511666353
  * [Wen05] Holger Wendland  “Scattered data approximation” 17, Cambridge Monographs on Applied and Computational Mathematics  Cambridge University Press, Cambridge, 2005, pp. x+336