# LAN property for discretely observed solutions to Lévy driven SDE’s

D. O. Ivanenko  Kyiv National Taras Shevchenko University, Volodymyrska, 64,
Kyiv, 01033, Ukraine<EMAIL_ADDRESS>and  A. M. Kulik  Institute of
Mathematics, Ukrainian National Academy of Sciences, 01601 Tereshchenkivska,
3, Kyiv, Ukraine<EMAIL_ADDRESS>
###### Abstract.

The LAN property is proved in the statistical model based on discrete-time
observations of a solution to a Lévy driven SDE. The proof is based on a
general sufficient condition for a statistical model based on a discrete
observations of a Markov process to possess the LAN property, and involves
substantially the Malliavin calculus-based integral representations for
derivatives of log-likelihood of the model.

###### Key words and phrases:

LAN property, Likelihood function, Lévy driven SDE, Regular statistical
experiment

##  1\. Introduction

Consider stochastic equation of the form

(1) $dX_{t}^{\theta}=a_{\theta}(X_{t}^{\theta})dt+dZ_{t},$

where $a:\Theta\times\mathbb{R}\to\mathbb{R}$ is a measurable function,
$\Theta=(\theta_{1},\theta_{2})\in\mathbb{R}$ is a parametric set. For a given
$\theta\in\Theta$, assuming that the drift term $a_{\theta}$ satisfies the
standard local Lipschitz and linear growth conditions, Eq. (1) uniquely
defines a Markov process $X$. The aim of this paper is to establish the _local
asymptotic normality_ property (LAN in the sequel) in a model, where the
process $X$ is discretely observed with a fixed time discretization value
$h>0$, and a number of observation $n\to\infty$.

The LAN property provides a convenient and powerful tool for establishing
lower efficiency bounds in a statistical model, e.g. [6], [17], [18]. Such a
property for statistical models, based on discrete observations of processes
with Lévy noise, was studied mostly in the cases, where the likelihood
function (or, at least its “main part”) is explicit, in a sense, e.g. [1],
[2], [7], [12], [13]. In the above references the models are linear in the
sense that the process under the observation is either a Lévy process, or a
solution of a linear (Ornstein-Uhlenbeck type) SDE driven by a Lévy process.
The general non-linear case remains non-studied to a great extent, and
apparently the main reason for this is that the transition probability density
of the observed Markov process in that case is highly implicit. In this paper
we develop tools, convenient for proving the LAN property in the framework of
discretely observed solutions to SDE’s with a Lévy noise. To make the
exposition reasonably transparent, we confine ourselves to a particular case
of one-dimensional and one-parameter model, and a fixed sample frequency $h$.
Various extensions (general state space, multiparameter model, high frequency
sampling, etc.) are visible, but we postpone their detailed analysis for a
further research.

Our approach consists of two principal parts. On one hand, we design a general
sufficient condition for a statistical model based on a discrete observations
of a Markov process to possess the LAN property, see Theorem 1 below. This
result extends the classical LeCam’s result about the LAN property for i.i.d.
samples, and it close [5, Theorem 13], with some substantial differences in
the basic assumptions, which makes our result well designed to a study of a
model based on observations of a Lévy driven SDE, see Remark 1 below. On the
other hand, we integral representations of derivatives of 1st and 2nd orders
of the log-likelihood are available: our recent papers [11] and [10] we have
derived such representations using the Malliavin calculus tools. The virtue of
this approach is the same with the one developed in [4] in the diffusion
setting, but with substantial changes which comes from non-diffusive structure
of the noise. Combination of these two principal parts leads to a required LAN
property.

The structure of the paper follows the two-stage scheme outlined above. First
we formulate in Section 2.1 (and prove in Section 3) a general sufficient
condition for the LAN property in a Markov model. Then we formulate in Section
2.2 (and prove in Section 4) our main result about the LAN property for a
discretely observed solution to a Lévy driven SDE; here the proof involves
substantially the Malliavin calculus-based integral representations of
derivatives of the log-likelihood from [11] and [10].

##  2\. The main results

###  2.1. LAN property for discretely observed Markov processes

Let $X$ be a Markov process taking its values in a locally compact metric
space $\mathbb{X}$. The law of $X$ is assumed to depend on a real-valued
parameter $\theta$; in what follows, we assume that the parametric set
$\Theta$ is an interval $(\theta_{1},\theta_{2})\in\mathbb{R}$. We denote by
$\mathsf{P}_{x}^{\theta}$ the law of $X$ with $X_{0}=x$, which corresponds to
the parameter value $\theta$; the expectation w.r.t. $\mathsf{P}_{x}^{\theta}$
is denoted by $\mathsf{E}_{x}^{\theta}$. For a given $h>0$, we denote by
$\mathsf{P}_{x,n}^{\theta}$ the law w.r.t. $\mathsf{P}_{x}^{\theta}$ of the
vector $X^{n}=\left\\{X_{hk},k=1,\dots,n\right\\}$ of discrete time
observations of $X$ with the step $h$. Denote by $\mathcal{E}_{n}$ the
statistical experiment generated by the sample $X^{n}$ with $X_{0}=x,$ i.e.

(2)
$\mathcal{E}_{n}=\Big{(}\mathbb{X}^{n},\mathcal{B}(\mathbb{X}^{n}),\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\Big{)};$

we refer to [8] for the notation and terminology. Our aim is to establish the
LAN property for the sequence of experiments $\\{\mathcal{E}_{n}\\}$.

Recall that the sequence of statistical experiments $\\{\mathcal{E}_{n}\\}$
(or, equivalently, the family
$\\{\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\\}$) is said to have _the LAN
property_ at the point $\theta_{0}\in\Theta$ as $n\rightarrow\infty$, if for
some sequence $r(n)>0,n\geq 1$ and all $u\in\mathbb{R}$

$Z_{n,\theta_{0}}(u):=\frac{d\mathsf{P}_{x,n}^{\theta_{0}+r(n)u}}{d\mathsf{P}_{x,n}^{\theta_{0}}}(X^{n})=\exp\left\\{\Delta_{n}(\theta_{0})u-\frac{1}{2}u^{2}+\Psi_{n}(u,\theta_{0})\right\\},$

with

(3) $\mathcal{L}\left(\Delta_{n}(\theta_{0})\ |\
\mathsf{P}_{x,n}^{\theta_{0}}\right)\Rightarrow N(0,1),\quad
n\rightarrow\infty;$ (4)
$\Psi_{n}(u,\theta_{0})\stackrel{{\scriptstyle\mathsf{P}_{x,n}^{\theta_{0}}}}{{\longrightarrow}}0,\quad
n\rightarrow\infty.$

In what follows we assume that $X$ admits a transition probability density
$p_{h}(\theta;x,y)$ w.r.t. some $\sigma$-finite measure $\lambda$.
Furthermore, we assume that the experiment $\mathcal{E}_{1}$ is _regular_ ;
that is, for every $x\in\mathbb{X}$

  * (a)

the function $\theta\mapsto p_{h}(\theta;x,y)$ is continuous for
$\lambda$-a.a. $y\in\mathbb{X}$;

  * (b)

the function $\sqrt{p_{h}(\theta;x,\cdot)}$ is differentiable in
$L_{2}(\mathbb{X},\lambda)$; that is, there exists $q_{h}(\theta;x,\cdot)\in
L_{2}(\mathbb{X},\lambda)$ such that

$\int_{\mathbb{X}}\left({\sqrt{p_{h}(\theta+\delta;x,y)}-\sqrt{p_{h}(\theta;x,y)}\over\delta}-q_{h}(\theta;x,y)\right)^{2}\lambda(dy)\to
0,\quad\delta\to 0;$

  * (c)

the function $q_{h}(\theta;x,\cdot)$ is continuous in
$L_{2}(\mathbb{X},\lambda)$ w.r.t. $\theta$; that is,

$\int_{\mathbb{X}}\Big{(}q_{h}(\theta+\delta;x,y)-q_{h}(\theta;x,y)\Big{)}^{2}\lambda(dy)\to
0,\quad\delta\to 0.$

Denote

(5) $g_{h}(\theta,x,y)=2q_{h}(\theta;x,y)\sqrt{p_{h}(\theta;x,y)};$

note that by the definition of $q_{h}$ the function $g_{h}$ is well defined
and satisfies

(6) $\mathsf{E}_{x}^{\theta}g_{h}(\theta;x,X_{h})=0$

for every $x\in\mathbb{R},\theta\in\Theta$. Furthermore, denote

(7)
$I_{n}(\theta)=\sum_{k=1}^{n}\mathsf{E}_{x}^{\theta}\Big{(}g_{h}(\theta;X_{h(k-1)},X_{hk})\Big{)}^{2}=4\mathsf{E}_{x}^{\theta}\sum_{k=1}^{n}\int_{\mathbb{X}}\left(q_{h}(\theta;X_{h(k-1)},y)\right)^{2}\lambda(dy).$

Assuming that the statistical experiment $\mathcal{E}_{n}$ is regular, the
above integral is finite and defines the _Fisher information_ for
$\mathcal{E}_{n}$.

We fix $\theta_{0}\in\Theta$, and put $r(n)=I_{n}^{-1/2}(\theta_{0})$ for $n$
large enough, assuming that for those $n$ one has $I_{n}(\theta_{0})>0$.

######  Theorem 1.

Suppose the following.

  * 1.

Statistical experiment (2) is regular for every $x\in\mathbb{X}$ and $n\geq
1$; for $n$ large enough $I_{n}(\theta_{0})>0$.

  * 2.

The sequence

$r(n)\sum_{j=1}^{n}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right),\quad n\geq
1$

is asymptotically normal w.r.t. $P_{x}^{\theta_{0}}$ with parameters $(0,1)$.

  * 3.

The sequence

$r^{2}(n)\sum_{j=1}^{n}g_{h}^{2}(\theta_{0};X_{h(j-1)},X_{hj}),\quad n\geq 1$

converges to 1 in $P_{x}^{\theta_{0}}$-probability.

  * 4.

There exists a constant $p>2$ such that

(8)
$\lim_{n\rightarrow\infty}r^{p}(n)\mathsf{E}_{x}^{\theta_{0}}\sum_{j=1}^{n}\left|g_{h}(\theta_{0};X_{h(j-1)},X_{hj})\right|^{p}=0.$

  * 5.

For every $N>0$

(9)
$\lim_{n\rightarrow\infty}\sup_{|v|<N}r^{2}(n)\mathsf{E}_{x}^{\theta_{0}}\sum_{j=1}^{n}\int_{\mathbb{X}}\left(q_{h}\left(\theta_{0}+r(n)v;X_{h(j-1)},y\right)-q_{h}(\theta_{0};X_{h(j-1)},y)\right)^{2}\lambda(dy)=0.$

Then $\\{\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\\}$ has the LAN property at
the point $\theta_{0}$.

######  Remark 1.

The above theorem is closely related to [5, Theorem 13]. One important
difference is that in [5] the main conditions are formulated in the terms of
the functions

$\sqrt{p_{h}(\theta+t;x,y)/p_{h}(\theta;x,y)}-1,$

while within our approach the main assumptions are imposed on the log-
likelihood derivative $g_{h}(\theta;x,y)$, and can be verified efficiently
e.g. in a model where $X$ is defined by an SDE with jumps; see Section 2.2
below. Another important difference is that the whole approach in [5] is
developed under the assumption that the log-likelihood function smoothly
depends on the parameter $\theta$. For a model where $X$ is defined by an SDE
with jumps, such an assumption may be very restrictive, see the detailed
discussion in [11]. This is the reason why we use instead the assumption of
regularity of the experiments, which both is much milder and is easily
verifiable, see [11].

Let us note briefly two possible extensions of the above result, which can be
obtained without any essential changes in the proof. We do not expose them
here in details, because they will not be used in the current paper.

######  Remark 2.

The statement of Theorem 1 still holds true if $h$ is allowed to depend on
$n$, with conditions 1 – 5 respectively changed.

######  Remark 3.

The statement of Theorem 1 still holds true if, instead of one $\theta_{0}$, a
sequence $\theta_{n}\to\theta_{0}$ is considered, with conditions 2 – 5
respectively changed. Moreover, in that case relation (3) and (4) would still
hold true if instead of a fixed $u$ a sequence $u_{n}\to u$ is considered.
That is, under the uniform version of conditions 2 – 5 the _uniform asymptotic
normality_ would hold true; see [8, Definition 2.2].

###  2.2. LAN property for families of distributions of solutions to Lévy
driven SDE’s

We assume that $Z$ in the SDE (1) is a Lévy process without a diffusion
component; that is,

$Z_{t}=ct+\int_{0}^{t}\int_{|u|>1}u\nu(ds,du)+\int_{0}^{t}\int_{|u|\leq
1}u\tilde{\nu}(ds,du),$

where $\nu$ is a Poisson point measure with the intensity measure $ds\mu(du)$,
and $\tilde{\nu}(ds,du)=\nu(ds,du)-ds\mu(du)$ is respective compensated
Poisson measure. In the sequel, we assume the Lévy measure $\mu$ to satisfy
the following.

H. (i) for some $\beta>0$,

$\int_{|u|\geq 1}u^{4+\beta}\mu(du)<\infty;$

(ii) for some $u_{0}>0$, the restriction of $\mu$ on $[-u_{0},u_{0}]$ has a
positive density $m\in
C^{2}\left(\left[-u_{0},0\right)\cup\left(0,u_{0}\right]\right)$;

(iii) there exists $C_{0}$ such that

$|m^{\prime}(u)|\leq C_{0}|u|^{-1}m(u),\quad|m^{\prime\prime}(u)|\leq
C_{0}u^{-2}m(u),\quad|u|\in(0,u_{0}];$

(iv)

$\left(\log\frac{1}{\varepsilon}\right)^{-1}\mu\Big{(}\\{u:|u|\geq\varepsilon\\}\Big{)}\to\infty,\quad\varepsilon\to
0.$

One particularly important class of Lévy processes satisfying H consists of
_tempered $\alpha$-stable processes_ (see [21]), which arise naturally in
models of turbulence [20], economical models of stochastic volatility [3],
etc.

Denote by $C^{k,m}(\mathbb{R}\times\Theta),k,m\geq 0$ the class of functions
$f:\mathbb{R}\times\Theta\to\mathbb{R}$ which has continuous derivatives

$\frac{\partial^{i}}{\partial x^{i}}\frac{\partial^{j}}{\partial\
\theta^{j}}f,\quad i\leq k,\quad j\leq m.$

About the coefficient $a_{\theta}(x)$ in Eq. (1) we assume the following.

A. (i) $a\in C^{3,2}(\mathbb{R}\times\Theta)$ have bounded derivatives
$\partial_{x}a$, $\partial^{2}_{xx}a$, $\partial^{2}_{x\theta}a$,
$\partial^{3}_{xxx}a$, $\partial^{3}_{x\theta\theta}a$,
$\partial^{3}_{xx\theta}a$, $\partial^{4}_{xxx\theta}a$ and

(10)
$|a_{\theta}(x)|+|\partial_{\theta}a_{\theta}(x)|+|\partial^{2}_{\theta\theta}a_{\theta}(x)|\leq
C(1+|x|),\quad\theta\in\Theta,\quad x\in\mathbb{R}.$

(ii) For a given $\theta_{0}\in\Theta$, there exists a neighbourhood
$(\theta_{-},\theta_{+})\subset\Theta$ of $\theta_{0}$ such that

$\limsup_{|x|\rightarrow\infty}\frac{a_{\theta}(x)}{x}<0\quad\hbox{uniformly
by }\theta\in(\theta_{-},\theta_{+}).$

It is proved in [11] that, under conditions A(i) and H, the following
properties hold:

  * •

the Markov process $X$ given by (1) has a transition probability density
$p_{t}^{\theta}$ w.r.t. the Lebesgue measure;

  * •

this density has a derivative $\partial_{\theta}p_{t}^{\theta}(x,y),$ and the
statistical experiment (2) is regular;

  * •

the function $g_{t}^{\theta}$, given by (5) satisfies (6).

Hence all the pre-requisites for Theorem 1, given in Section 2.1, are
available with $\lambda(dx)=dx$ (the Lebesgue measure).

Furthermore, under conditions A and H, for $\theta=\theta_{0}$ corresponding
Markov process $X$ is ergodic, i.e. there exists unique invariant probability
measure $\varkappa_{inv}^{\theta_{0}}$ for $X$. One can verify this easily,
using conditions, sufficient for ergodicity of solutions to Lévy driven SDE’s,
given in [19] and [14]. Denote by $\\{X^{st,\theta_{0}}_{t},t\in\mathbb{R}\\}$
corresponding stationary version of $X$; that is, a Markov process, defined on
whole axis $\mathbb{R}$, which has the transition probabilities with $X$ and
one-dimensional distributions equal to $\varkappa_{inv}^{\theta_{0}}$.
Clearly, the existence of such a process, on a proper probability space, is
guaranteed by the Kolmogorov consistency theorem. Denote

(11)
$\sigma^{2}(\theta_{0})=\mathsf{E}\Big{(}g_{h}(\theta_{0};X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}})\Big{)}^{2}=\int_{\mathbb{R}}\int_{\mathbb{R}}\Big{(}g_{h}(\theta_{0};x,y)\Big{)}^{2}p_{h}(\theta_{0};x,y)\,dy\varkappa_{inv}^{\theta_{0}}(dx).$

The following theorem performs the main result of this paper. Its proof is
given in Section 4 below.

######  Theorem 2.

Let conditions A and H hold true, and

$\sigma^{2}(\theta_{0})>0.$

Then the family $\\{\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\\}$ possesses
the LAN property at the point $\theta=\theta_{0}$.

##  3\. Proof of Theorem 1

The method of proof goes back to LeCam’s proof of the LAN property for i.i.d.
samples, see e.g. Theorem II.1.1 and Theorem II.3.1 in [8]. In the Markov
setting, the dependence in the observations lead to some additional
technicalities; see e.g. (19). The possible ways to overcome these additional
difficulties can be found, in a slightly different setting, in the proof of
[5, Theorem 13]. In order to keep the exposition transparent and self-
sufficient, we prefer to give a complete proof of Theorem 1 explicitly, rather
than to give a chain of partly relevant references.

We separate the proof into several lemmas; in all the lemmas in this section
we assume the conditions of Theorem 1 to be fulfilled. Values $x,\theta_{0},$
and $u$ are fixed; we assume that $n$ is large enough, so that
$\theta_{0}+r(n)u\in\Theta$. In order to simplify the notation below we write
$\theta$ instead of $\theta_{0}$.

Denote

$\zeta^{\theta}_{jn}(u)=\left(\left(\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}\right)^{1/2}-1\right)I\left\\{p_{h}(\theta;X_{h(j-1)},X_{hj})\neq
0\right\\}.$

######  Lemma 1.

One has

(12)
$\limsup_{n\rightarrow\infty}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}\leq\frac{1}{4}u^{2}$

and

(13)
$\lim_{n\rightarrow\infty}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}=0.$

###### Proof.

By the regularity of $\mathcal{E}_{1}$ and the Cauchy inequality we have

$\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}\\\
=\mathsf{E}_{x}^{\theta}\int\limits_{\\{y:p^{\theta}_{h}(z,y)\neq
0\\}}\left(\sqrt{p_{h}\left(\theta+r(n)u;X_{h(j-1)},y\right)}\right.\\\
\left.-\sqrt{p_{h}\left(\theta;X_{h(j-1)},y\right)}-r(n)uq_{h}(\theta;X_{h(j-1)},y)\right)^{2}\lambda(dy)\\\
\leq(r(n)u)^{2}\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\left(\int_{0}^{1}q_{h}\left(\theta+r(n)uv,X_{h(j-1)},y\right)-q_{h}\left(\theta;X_{h(j-1)},y\right)dv\right)^{2}\lambda(dy)\\\
\leq(r(n)u)^{2}\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\lambda(dy)\int_{0}^{1}\left(q_{h}\left(\theta+r(n)uv;X_{h(j-1)},y\right)-q_{h}\left(\theta;X_{h(j-1)},y\right)\right)^{2}dv.$

This and (9) yield (13). To deduce (12) from (13), recall an elementary
inequality

(14) $|AB|\leq\frac{\alpha}{2}A^{2}+\frac{1}{2\alpha}B^{2},\quad\alpha>0,$

and write

$\zeta^{\theta}_{jn}(u)=\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})+\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)=:A+B.$

Then

$\displaystyle\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}$
$\displaystyle\leq(1+\alpha){1\over
4}u^{2}r^{2}(n)\mathsf{E}_{x}^{\theta}\left(g_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}$
$\displaystyle+\left(1+{1\over\alpha}\right)\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}.$

Because by the construction

(15)
$\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(g_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}=I_{n}(\theta)=r^{-2}(n),$

this leads to the bound

$\limsup_{n\rightarrow\infty}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}\leq\frac{1+\alpha}{4}u^{2}.$

Since $\alpha>0$ is arbitrary, this completes the proof. ∎

######  Lemma 2.

One has

(16) $\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}\to\frac{u^{2}}{4},\quad
n\to\infty$

in $\mathsf{P}_{x}^{\theta}$-probability.

###### Proof.

By the Chebyshev inequality,

$\mathsf{P}_{x}^{\theta}\left\\{\left|\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}-\frac{1}{4}r^{2}(n)u^{2}\sum_{j=1}^{n}(g_{h}(\theta;X_{h(j-1)},X_{hj}))^{2}\right|>\varepsilon\right\\}\\\
\leq\frac{1}{\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left|(\zeta^{\theta}_{jn}(u))^{2}-\frac{1}{4}r^{2}(n)u^{2}(g_{h}(\theta;X_{h(j-1)},X_{hj}))^{2}\right|\\\
=\frac{1}{\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left|\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right|\left|\zeta^{\theta}_{jn}(u)+\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right|$

which by (14), for a given $\alpha>0$, is dominated by

$\frac{1}{2\alpha\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}\\\
+\frac{\alpha}{2\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)+\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}.$

By (13) the first item of this expression tends to zero as
$n\rightarrow\infty$. Furthermore, the Cauchy inequality together with (12)
and (15) imply that for the second one the upper limit does not exceed

$\limsup_{n\to\infty}\left(\frac{\alpha}{\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}+\frac{\alpha
u^{2}}{2\varepsilon}r^{2}(n)\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(g_{h}\left(\theta;X_{h(j-1)},X_{hj}\right))^{2}\right)\leq\frac{3\alpha
u^{2}}{2\varepsilon}.$

Since $\alpha>0$ is arbitrary, this proves that the difference

$\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}-\frac{1}{4}r^{2}(n)u^{2}\sum_{j=1}^{n}(g_{h}(\theta;X_{h(j-1)},X_{hj}))^{2}$

tends to $0$ in $\mathsf{P}_{x}^{\theta}$-probability. Combined with the
condition 3 of Theorem 1, this gives the required statement. ∎

######  Lemma 3.

One has

(17) $\max_{1\leq j\leq n}|\zeta^{\theta}_{jn}(u)|\to 0,\quad n\to\infty$

in $\mathsf{P}_{x}^{\theta}$-probability.

###### Proof.

We have

$\mathsf{P}_{x}^{\theta}\left\\{\max_{1\leq j\leq
n}|\zeta^{\theta}_{jn}(u)|>\varepsilon\right\\}\leq\sum_{j=1}^{n}\mathsf{P}_{x}^{\theta}\left\\{|\zeta^{\theta}_{jn}(u)|>\varepsilon\right\\}\\\
\leq\sum_{j=1}^{n}\mathsf{P}_{x}^{\theta}\left\\{\left|\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right|>\frac{\varepsilon}{2}\right\\}\\\
+\sum_{j=1}^{n}\mathsf{P}_{x}^{\theta}\left\\{\left|g_{h}(\theta;X_{h(j-1)},X_{hj})\right|>\frac{\varepsilon}{4r(n)|u|}\right\\}.$

The first sum in the r.h.s. of this inequality vanishes as
$n\rightarrow\infty$ because of (13), the second sum vanishes because of the
condition 4 of Theorem 1. ∎

######  Corollary 1.

By Lemma 3 and Lemma 2, we have

(18) $\sum_{j=1}^{n}|\zeta^{\theta}_{jn}(u)|^{3}\to 0,\quad n\to\infty$

in $\mathsf{P}_{x}^{\theta}$-probability.

Because of the Markov structure of the sample, in addition to Lemma 2 we will
need the following statement. Denote

$\mathcal{F}_{j}=\sigma(X_{hi},i\leq
j),\quad\mathsf{E}_{x,j}^{\theta}=\mathsf{E}_{x}^{\theta}[\cdot|\mathcal{F}_{j}].$

######  Lemma 4.

One has

(19)
$\sum_{j=1}^{n}\mathsf{E}_{x,j-1}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}\to\frac{u^{2}}{4},\quad
n\to\infty$

in $\mathsf{P}_{x}^{\theta}$-probability.

###### Proof.

Denote

$\chi_{jn}=(\zeta^{\theta}_{jn}(u))^{2}-\mathsf{E}_{x,j-1}^{\theta}(\zeta^{\theta}_{jn}(u))^{2},\quad
S_{n}=\sum_{j=1}^{n}\chi_{jn},$

then by (16) it us enough to prove that $S_{n}\to 0$ in
$\mathsf{P}_{x}^{\theta}$-probability. Fix $\varepsilon>0$, and put

$\chi_{jn}^{\varepsilon}=(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|\leq\varepsilon}-\mathsf{E}_{x,j-1}^{\theta}\Big{(}(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|\leq\varepsilon}\Big{)},\quad
S_{n}^{\varepsilon}=\sum_{j=1}^{n}\chi_{jn}^{\varepsilon}.$

By the construction $\\{\chi_{j}^{\varepsilon},j=1,\dots,n\\}$ is a martingale
difference, hence

$\displaystyle\mathsf{E}_{x}^{\theta}(S_{n}^{\varepsilon})^{2}$
$\displaystyle=\sum_{k=1}^{n}\mathsf{E}_{x}^{\theta}(\chi_{jn}^{\varepsilon})^{2}\leq\sum_{k=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{4}1_{|\zeta^{\theta}_{jn}(u)|\leq\varepsilon}\leq\varepsilon^{2}\mathsf{E}_{x}^{\theta}\sum_{k=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}.$

Hence by (12) and the Cauchy inequality,

(20)
$\limsup_{n\to\infty}\mathsf{E}_{x}^{\theta}|S_{n}^{\varepsilon}|\leq{\varepsilon|u|\over
2}$

Now, let us estimate the difference $S_{n}-S_{n}^{\varepsilon}$. Note that,
using the first statement in Lemma 1, one can improve the statement of Lemma 2
and show that the convergence (16) holds true in
$L_{1}(\mathsf{P}_{x}^{\theta})$; see e.g. Theorem A.I.4 in [8]. In
particular, this means that the sequence

$\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2},\quad n\geq 1$

is uniformly integrable. Hence, because by Lemma 3 the probabilities of the
sets

(21) $\Omega_{n}^{\varepsilon}=\left\\{\max_{j\leq
n}|\zeta_{jn}|>\varepsilon\right\\}$

tend to zero as $n\to\infty$, we have

$\mathsf{E}_{x}^{\theta}\left(1_{\Omega_{n}^{\varepsilon}}\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}\right)\to
0.$

One has

$\chi_{jn}-\chi_{jn}^{\varepsilon}=(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|>\varepsilon}-\mathsf{E}_{x,j}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|>\varepsilon},$

hence

$\displaystyle\mathsf{E}_{x}^{\theta}|S_{n}-S_{n}^{\varepsilon}|$
$\displaystyle\leq
2\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|>\varepsilon}\leq
2\mathsf{E}_{x}^{\theta}\left(1_{\Omega_{n}^{\varepsilon}}\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}\right)\to
0.$

Together with (20) this gives

$\limsup_{n\to\infty}\mathsf{E}_{x}^{\theta}|S_{n}|\leq{\varepsilon|u|\over
2},$

which completes the proof because $\varepsilon>0$ is arbitrary. ∎

The final preparatory result we require is the following.

######  Lemma 5.

One has

(22)
$2\sum_{j=1}^{n}\zeta^{\theta}_{jn}(u)-r(n)u\sum_{j=1}^{n}g_{h}(\theta;X_{h(j-1)},X_{hj})\to-\frac{u^{2}}{4},\quad
n\to\infty$

in $\mathsf{P}_{x}^{\theta}$-probability.

###### Proof.

We have the equality

$(\zeta^{\theta}_{jn}(u))^{2}=\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}-1-2\zeta^{\theta}_{jn}(u)$

valid $\mathsf{P}_{x}^{\theta}$-a.s. Note that by the Markov property of $X$
one has

$\displaystyle\mathsf{E}_{x,j-1}^{\theta}$
$\displaystyle\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}$
$\displaystyle=\int_{\mathbb{X}}\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},y)}{p_{h}(\theta;X_{h(j-1)},y)}p_{h}(\theta;X_{h(j-1)},y)\lambda(dy)=1;$

hence by Lemma 4 one has that

$\sum\limits_{j=1}^{n}\mathsf{E}_{x,j-1}^{\theta}\zeta^{\theta}_{jn}(u)\to-\frac{u^{2}}{8}$

in $\mathsf{P}_{x}^{\theta}$-probability. Therefore, what we have to prove in
fact is that

$V_{n}:=2\sum_{j=1}^{n}\left(\zeta^{\theta}_{jn}(u)-\mathsf{E}_{x,j-1}^{\theta}\zeta^{\theta}_{jn}(u)\right)-r(n)u\sum_{j=1}^{n}g_{h}(\theta;X_{h(j-1)},X_{hj})\to
0$

in $\mathsf{P}_{x}^{\theta}$-probability. By (6) the sequence

$\zeta^{\theta}_{jn}(u)-\mathsf{E}_{x,j-1}^{\theta}\zeta^{\theta}_{jn}(u)-r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj}),\quad
j=1,\dots n$

is a martingale difference, hence

$\mathsf{E}_{x}^{\theta}V_{n}^{2}\leq
4\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2},$

which tends to zero as $n\to\infty$ by (13). ∎

Now, we can finalize the proof of Theorem 1. Fix $\varepsilon\in(0,1)$ and
consider the sets $\Omega_{n}^{\varepsilon}$ defined by (21); by Lemma 3 we
have $\mathsf{P}_{x}^{\theta}(\Omega_{n}^{\varepsilon})\to 0$. Using the
Taylor expansion for the function $\log(1+x)$, we obtain that there exist a
constant $C_{\varepsilon}$ and random variables $\alpha_{jn}$ such that
$|\alpha_{jn}|<C_{\varepsilon}$, for which the following identity holds true
outside of the set $\Omega_{n}^{\varepsilon}$:

$\sum_{j=1}^{n}\log\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}=2\sum_{j=1}^{n}\zeta^{\theta}_{jn}(u)-\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}+\sum_{j=1}^{n}\alpha_{jn}|\zeta^{\theta}_{jn}(u)|^{3}.$

Then by Lemma 2, Lemma 5, and Corollary 1 we have

$\displaystyle\log Z_{n,\theta}(u)$
$\displaystyle=\sum_{j=1}^{n}\log\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}$
$\displaystyle\hskip
28.45274pt=r(n)u\sum_{j=1}^{n}g_{h}(\theta;X_{h(j-1)},X_{hj})-\frac{u^{2}}{4}-\frac{u^{2}}{4}+\Psi_{n},$

where $\Psi_{n}\to 0$ in $\mathsf{P}_{x}^{\theta}$-probability. By the
asymptotic normality condition 2, this completes the proof. ∎

##  4\. Proof of Theorem 2

To prove Theorem 2, we verify the conditions of Theorem 1. First, let us give
an auxiliary result, which will be used repeatedly in the proof.

######  Lemma 6.

Under conditions A and H for every $p\in(2,4+\beta)$ there exists a constant
$C$ such that for all $x\in\mathbb{R}$, $\theta\in(\theta_{-},\theta_{+})$,
and $t\geq 0$

(23) $\mathsf{E}_{x}^{\theta}\Big{|}g_{h}(\theta;x,X_{h})\Big{|}^{p}\leq
C(1+|x|)^{p},\quad\mathsf{E}_{x}^{\theta}|X_{t}|^{p}\leq C(1+|x|^{p}).$

###### Proof.

The first inequality is proved in Lemma 1 [11]. One can prove the second
inequality, using a standard argument based on the Lyapunov condition for the
function $V(x)=|x|^{p};$ e.g. Proposition 4.1 [14]. ∎

Recall (e.g. [14], Section 3.2) that one standard way to construct the
invariant measure $\varkappa^{\theta_{0}}_{inv}$ is to take a weak limit point
(as $T\to\infty$) for the family of _Khas’minskii’s averages_

$\varkappa^{\theta_{0}}_{T}(dy)={1\over
T}\int_{0}^{T}\mathsf{P}_{x}^{\theta_{0}}(X_{t}\in dy)\,dt.$

Then, by the Fatou lemma, the second relation in (23) implies the following
moment bound for $\varkappa^{\theta_{0}}_{inv}$.

######  Corollary 2.

For every $p\in(2,4+\beta)$,

$\int_{\mathbb{R}}|y|^{p}\varkappa^{\theta_{0}}_{inv}(dy)<\infty.$

Everywhere below we assume conditions of Theorem 2 to hold true.

######  Lemma 7.

The sequence

${1\over\sqrt{n}}\sum_{j=1}^{n}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right),\quad
n\geq 1$

is asymptotically normal w.r.t. $P_{x}^{\theta_{0}}$ with parameters
$(0,\sigma^{2}(\theta_{0}))$, where $\sigma^{2}(\theta_{0})$ is defined in
(11).

###### Proof.

The idea of the proof is similar to the one of the proof of Theorem 3.3 [16].
Denote

$Q_{n}(\theta_{0},X)={1\over\sqrt{n}}\sum_{j=1}^{n}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right).$

By Theorem 2.2 [19] (see also Theorem 1.2 [14]), the $\alpha$-mixing
coefficient $\alpha(t)$ for the stationary version of the process $X$ does not
exceed $C_{3}e^{-C_{4}t}$, where $C_{3},C_{4}$ are some positive constants.
Then by CLT for stationary sequences (Theorem 18.5.3 [9]) and (23) we have

$Q_{n}(\theta_{0},X^{st,\theta_{0}})\Rightarrow\mathcal{N}\left(0,\widetilde{\sigma}^{2}(\theta_{0})\right),\quad
n\rightarrow\infty$

with

$\widetilde{\sigma}^{2}(\theta_{0})=\sum_{k=-\infty}^{+\infty}\mathsf{E}\left(g_{h}\left(\theta_{0};X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}}\right)g_{h}\left(\theta;X_{h(k-1)}^{st,\theta_{0}},X_{hk}^{st,\theta_{0}}\right)\right).$

Furthermore, under conditions of Theorem 2 there exists an exponential
coupling for the process $X$; that is, a two-component process
$Y=(Y^{1},Y^{2})$, possibly defined on another probability space, such that
$Y^{1}$ has the distribution $\mathsf{P}_{x}^{\theta_{0}}$, $Y^{2}$ has the
same distribution with $X^{st,\theta_{0}}$, and for all $t>0$

(24) $\mathsf{P}\Big{(}Y_{t}^{1}\neq Y_{t}^{2}\Big{)}\leq C_{1}e^{-C_{2}t}$

with some constants $C_{1}$, $C_{2}$. The proof of this fact can be found in
[15] (Theorem 2.2). Then for any Lipschitz continuous function
$f:\mathbb{R}\to\mathbb{R}$ we have

(25)
$|\mathsf{E}_{x}^{\theta}f(Q_{n}(\theta_{0},X))-\mathsf{E}f(Q_{n}(\theta_{0},X^{st,\theta_{0}}))|=|\mathsf{E}f(Q_{n}(\theta_{0},Y^{1}))-\mathsf{E}f(Q_{n}(\theta_{0},Y^{2}))|\\\
=\mathrm{Lip}(f)\mathsf{E}|Q_{n}(\theta_{0},Y^{1})-Q_{n}(\theta_{0},Y^{2})|\\\
\leq\frac{\mathrm{Lip}(f)}{\sqrt{n}}\sum_{k=1}^{n}\mathsf{E}\left|g_{h}(\theta_{0};Y^{1}_{h(k-1)},Y^{1}_{hk})-g_{h}(\theta_{0};Y^{2}_{h(k-1)},Y^{2}_{hk})\right|1_{(Y^{1}_{h(k-1)},Y^{1}_{hk})\neq(Y^{2}_{h(k-1)},Y^{2}_{hk})}\\\
\leq\frac{2\mathrm{Lip}(f)}{\sqrt{n}}\sum_{k=1}^{n}\left(\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{1}_{h(k-1)},Y^{1}_{hk}\right)\right|^{p}+\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{2}_{h(k-1)},Y^{2}_{hk}\right)\right|^{p}\right)^{1/p}\\\
\times\left(P\Big{(}Y^{1}_{h(k-1)}\neq
Y^{2}_{h(k-1)}\Big{)}+P\Big{(}Y^{1}_{hk}\neq Y^{2}_{hk}\Big{)}\right)^{1/q},$

where $p,q>1$ are such that $1/p+1/q=1$. Since $Y^{1}$ has the distribution
$\mathsf{P}_{x}^{\theta_{0}}$, by (23) we have for $p\in n(2,4+\beta)$

(26)
$\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{1}_{h(k-1)},Y^{1}_{hk}\right)\right|^{p}=\mathsf{E}_{x}^{\theta_{0}}\left|g_{h}\left(\theta_{0};X_{h(k-1)},X_{hk}\right)\right|^{p}\\\
\leq C\mathsf{E}_{x}^{\theta_{0}}\Big{(}1+|X_{h(k-1)}|^{p})\Big{)}\leq
C+C^{2}(1+|x|^{p}).$

Similarly,

(27)
$\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{2}_{h(k-1)},Y^{2}_{hk}\right)\right|^{p}=\mathsf{E}\left|g_{h}\left(\theta_{0};X_{h(k-1)}^{st,\theta_{0}},X_{hk}^{st,\theta_{0}}\right)\right|^{p}\\\
\leq
C\mathsf{E}\Big{(}1+\Big{|}X_{h(k-1)}^{st,\theta_{0}}\Big{|}^{p})\Big{)}=C+C\int_{\mathbb{R}}|y|^{p}\varkappa^{\theta_{0}}_{inv}(dy),$

and the constant in the right hand side is finite by Corollary 2. Hence (24)
and (25) yield that

$\mathsf{E}_{x}^{\theta}f(Q_{n}(\theta_{0},X))\to\mathsf{E}f(\xi),\quad
n\to\infty,\quad\xi\sim\mathcal{N}\left(0,\widetilde{\sigma}^{2}(\theta_{0})\right)$

for every Lipschitz continuous function $f:\mathbb{R}\to\mathbb{R}$. This
means that the sequence $Q_{n}(\theta_{0},X),n\geq 1$ is asymptotically normal
w.r.t. $P_{x}^{\theta_{0}}$ with parameters
$(0,\tilde{\sigma}^{2}(\theta_{0}))$.

To conclude the proof, it remains to show that
$\widetilde{\sigma}^{2}(\theta_{0})={\sigma}^{2}(\theta_{0})$. This follows
easily from (6) because, by the Markov property of $X^{st,\theta_{0}}$,

$\widetilde{\sigma}^{2}(\theta_{0})=\sigma^{2}(\theta_{0})+2\sum_{k=1}^{\infty}\mathsf{E}\left(g_{h}\left(\theta_{0};X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}}\right)g_{h}\left(\theta;X_{h(k-1)}^{st,\theta_{0}},X_{hk}^{st,\theta_{0}}\right)\right)\\\
=\sigma^{2}(\theta_{0})+2\sum_{k=1}^{\infty}\mathsf{E}\left[g_{h}(\theta;X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}})\Big{(}\mathsf{E}_{x}^{\theta}g_{h}(\theta_{0};x,X_{h})\Big{)}_{x=X_{h(k-1)}^{st,\theta_{0}}}\right].$

∎

Similarly, one can prove that

${1\over{n}}\sum_{j=1}^{n}\Big{(}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right)\Big{)}^{2}\to\sigma^{2}(\theta_{0}),\quad
n\to\infty$

in $L_{1}(\mathsf{P}_{x}^{\theta_{0}})$; the argument is completely the same,
with the CLT for a stationary sequence replaced by the Birkhoff-Khinchin
ergodic theorem (we omit the details). Hence

(28) $I_{n}(\theta_{0})\sim n\sigma^{2}(\theta_{0}),\quad
r(n)\sim{1\over\sqrt{n}\sigma(\theta_{0})},\quad n\to\infty.$

Therefore conditions 2 – 4 of Theorem 1 are verified. Condition 1 of Theorem 1
also holds true: regularity property is proved in [11], and positivity of
$I_{n}(\theta)$ follows from (28).

Let us prove (9), which then would allow us to apply Theorem 1. It is proved
in [10] that, under the conditions of Theorem 2, the function
$q_{h}(\theta,x,y)$ is $L_{2}$-differentiable w.r.t. $\theta$, and

$\partial_{\theta}q_{h}=\frac{1}{2}(\partial_{\theta}g_{h})\sqrt{p_{h}}+\frac{1}{4}(g_{h})^{2}\sqrt{p_{h}}.$

In addition, it is proved therein that for every $\gamma\in[1,2+\beta/2)$

(29)
$\mathsf{E}_{x}^{\theta}\Big{|}\partial_{\theta}g_{h}(\theta;x,X_{h})\Big{|}^{\gamma}\leq
C(1+|x|)^{\gamma}.$

Then

$\displaystyle\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\left(q_{h}\left(\theta+r(n)v,X_{h(j-1)},y\right)-q_{h}(\theta,X_{h(j-1)},y)\right)^{2}dy$
$\displaystyle\hskip 56.9055pt\leq
r(n)v\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}dy\int_{0}^{r(n)v}\left(\partial_{\theta}q_{h}\left(\theta+s,X_{h(j-1)},y\right)\right)^{2}ds$
$\displaystyle\hskip
56.9055pt\leq\frac{r(n)v}{4}\mathsf{E}_{x}^{\theta}\int_{0}^{r(n)v}ds\int_{\mathbb{R}}\left(\partial_{\theta}g_{h}\left(\theta+s;X_{h(j-1)},y\right)+\frac{1}{2}g_{h}\left(\theta+s;X_{h(j-1)},y\right)^{2}\right)^{2}$
$\displaystyle\hskip 199.16928pt\times p_{h}^{s}(X_{h(j-1)},y)dy$
$\displaystyle\hskip 56.9055pt\leq
Cr(n)^{2}v^{2}\mathsf{E}_{x}^{\theta}\left(1+(X_{h(j-1)})^{4}\right);$

in the last inequality we have used (29) and the first relation in (23). Using
the second relation in (23), we get then

$\sup_{|v|<N}r(n)^{2}\mathsf{E}_{x}^{\theta}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\left(q_{h}\left(\theta+r(n)v,X_{h(j-1)},y\right)-q_{h}(\theta,X_{h(j-1)},y)\right)^{2}dy\leq
CN^{2}nr(n)^{4}$

with a constant $C$ that depends only on $x$. This relation together with (28)
completes the proof.

## Acknowledgements

The authors are deeply grateful to H. Masuda for a valuable bibliographic help
and useful discussion.

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11institutetext: Dipartimento di Matematica e Fisica "Ennio De Giorgi",
UniversitÃ del Salento and INFN-Lecce, Via Arnesano, 73100 Lecce, Italy

# Asymptotic symmetries and soft theorems in higher-dimensional gravity

Stefano Lionetti 11<EMAIL_ADDRESS>
###### Abstract

Soft theorems can be recast as Ward identities of asymptotic symmetries. We
review such relation for the leading and subleading soft graviton theorems in
arbitrary even dimensions. While soft theorems are trivially generalized to
dimensions higher than four, the charges of asymptotic symmetries are plagued
by divergences requiring a renormalization. We argue that the renormalized
charges of these symmetries can be determined by rewriting soft theorems as
Ward identities. In order to show that the charges of such identities generate
asymptotic symmetries, we propose a suitable commutation relation among
certain components of the metric fields

##  1 Introduction

Over the last few years a triangular equivalence relation was discovered
connecting three apparently different topics: asymptotic symmetries, soft
theorems and memory effects Strominger:2013jfa ; He:2014laa ;
Strominger:2014pwa . This equivalence relation can be drawn potentially in
every theory with a massless particle, for example in QED, QCD, SUSY and
gravity (for a review of the topic, see Strominger:2017zoo ). In this paper we
focus on the relationship between asymptotic symmetries and soft theorems at
leading and subleading order in gravity in arbitrary even dimensions.

In gravity the asymptotic symmetries are diffeomorphisms that preserve the
asymptotic flatness of spacetime Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza .
Such transformations map an asymptotically flat metric into an other
asymptotically flat metric. The asymptotic symmetry group is an infinite-
dimensional extension of the PoincarÃ group. The Ward identitity of asymptotic
symmetries turn out to be nothing but soft theorems.

The soft graviton theorem Weinberg:1965nx ; Cachazo:2014fwa  is a universal
formula relating scattering amplitudes that differ only by the addition of a
graviton whose energy $\omega$ is taken to zero

$\mathcal{M}_{n+n^{\prime}+1}\left(q=\omega\hat{q};p_{1},\ldots,p_{n+n^{\prime}}\right)=\left[S^{(1)}+S^{(2)}\right]\mathcal{M}_{n+n^{\prime}}\left(p_{1},\ldots,p_{n+n^{\prime}}\right)+\mathcal{O}\left(\omega\right)$
(1)

where $\\{p_{1},\ldots p_{n}\\}$ are the momenta of the incoming particles,
while $\\{p_{n+1},\ldots p_{n+n^{\prime}}\\}$ are the momenta of the outgoing
particles. $S^{(1)}$ and $S^{(2)}$ are given by

$\displaystyle S^{(1)}$
$\displaystyle\equiv\frac{\kappa}{2}\left(\sum_{k=n+1}^{n+n^{\prime}}\frac{\varepsilon_{\mu\nu}p_{k}^{\mu}p_{k}^{\nu}}{p_{k}\cdot
q}-\sum_{k=1}^{n}\frac{\varepsilon_{\mu\nu}p_{k}^{\mu}p_{k}^{\nu}}{p_{k}\cdot
q}\right)$ (2) $\displaystyle S^{(2)}$
$\displaystyle\equiv-\frac{i\kappa}{2}\left(\sum_{k=n+1}^{n+n^{\prime}}\frac{\varepsilon_{\mu\nu}p_{k}^{\mu}q_{\rho}J_{k}^{\rho\nu}}{p_{k}\cdot
q}-\sum_{k=1}^{n}\frac{\varepsilon_{\mu\nu}p_{k}^{\mu}q_{\rho}J_{k}^{\rho\nu}}{p_{k}\cdot
q}\right)$

where $\kappa^{2}=32\pi G$ and $J_{k}^{\rho\nu}$ is the total angular momentum
(orbital+spin) of the $k$-th particle and $\varepsilon_{\mu\nu}$ is the
polarization tensor of the graviton. In the soft limit $\omega\rightarrow 0$,
the leading term given by $S^{(1)}$ is of order $1/\omega$, while the
subleading term given by $S^{(2)}$ is constant in $\omega$. The formula indeed
comprises the leading and subleading soft graviton theorem. We are not
concerned with higher order terms in our analysis.

In four dimensions the leading soft theorem is equivalent to the
supertranslation Ward identity Strominger:2013jfa ; He:2014laa , while the
subleading soft theorem is equivalent to the Diff$(S^{2})$ Ward identity
Campiglia:2014yka ; Campiglia:2015yka . When computing the Diff$(S^{2})$
charges, we encounter divergences and we need to take care of them by adding
appropriate counterterms to the action as shown in Compere:2018ylh ;
Compere:2020lrt ; Chandrasekaran:2021vyu . As we will see, in $d>4$ such
divergences are even more present (supertranslations charges will require a
renormalization too). Together, supertranslations and Diff$(S^{2})$ make up
the asymptotic symmetry group as defined in Campiglia:2014yka ;
Campiglia:2015yka

$\mathcal{AG}=\operatorname{Diff}\left(S^{2}\right)\ltimes\text{
Supertranslations }$ (3)

The equivalence relation between asymptotic symmetries and soft theorems seems
to be somewhat more obscure in dimensions higher than four. Indeed soft
gravitons theorems hold in any dimension $d=2m+2$ while both supertranslations
and Diff$(S^{2m})$ charges seem to diverge in $d>4$. Many researchers avoided
such divergences’ problem by imposing strong falloff conditions, leaving only
the Poincaré group as part of the asymptotic symmetry group Hollands:2016oma ;
Hollands:2003ie ; Tanabe:2011es . In $d=4$, such argument would not hold.
Falloff conditions disallowing supertranslations and Diff$(S^{2})$
automatically exclude all the generic radiative solutions in $d=4$. In any
case, we will not consider such strong falloff conditions in $d>4$ either,
since our aim is to preserve the equivalence relation between asymptotic
symmetries and soft theorems. However, a renormalization seems to be mandatory
in order to solve the divergence problems.111In Aggarwal:2018ilg  the author
proposed boundary conditions that eliminate divergences in the
supertranslations charges. However these boundary conditions are only
consistent in linearized gravity and a renormalization is still needed in the
full theory.

In this paper we present the proper asymptotic symmetry group that can be
linked to soft graviton theorems. We then rewrite the leading and subleading
soft graviton theorems as Ward identities. We argue that such identities are
associated to asymptotic symmetries by proposing a suitable commutation
relation among certain components of the metric fields. We expect the need for
a renormalization in order to prove such commutation relation. We work in the
Bondi gauge and linearized gravity coupled to massless matter throughout the
paper. Our discussion is restricted to tree-level. Asymptotic symmetries have
also been studied in higher dimensions in the full nonlinear theory, for any
spin, in odd-dimensional cases and in asymptotically (A)dS spacetimes
Campoleoni:2020ejn ; Capone:2021ouo ; Fuentealba:2022yqt ; Bekaert:2022ipg ;
Fiorucci:2020xto ; Chowdhury:2022nus .

##  2 Asymptotic symmetries in higher dimensions

In gravity the asymptotic symmetries are diffeomorphisms that preserve the
asymptotic flatness of spacetime. In order to define precisely asymptotic
flatness, we need to specify the rate at which the metric approaches a
Minkowski metric at asymptotically large distances. Unfortunately there is no
unambiguous method of determining such asymptotic falloff conditions which are
often only a posteriori justified. There are however guidelines that must be
followed: the falloff conditions should be weak enough so that all interesting
solutions are allowed, but strong enough to rule out unphysical solutions,
such as those with infinite energy. Various options are discussed in the
literature leading to different results Hollands:2016oma ; Pate:2017fgt ;
Kapec:2015vwa ; Colferai:2020rte . In this paper, we advocate for falloff
conditions leading to non-trivial relations among $\mathcal{S}-$matrix
elements. Indeed Ward identities, associated to proper asymptotic symmetries,
turn out to be nothing but soft theorems.

We are interested in asymptotically flat spacetimes at both future and past
null infinity $\mathcal{I}^{\pm}$. For concreteness, let us focus on future
null infinity. We choose the coordinate system $(u,r,z^{a})$, where $u=t-r$ is
the retarded time, $r$ is the radial coordinate and $z^{a}$ ${(a=1,\dots,2m)}$
are the coordinates on the sphere $S^{2m}$. We work in the Bondi gauge
imposing the following $2m+2$ conditions

$g_{rr}=0,\quad\quad
g_{ra}=0,\quad\quad\operatorname{det}g_{ab}=r^{4m}\operatorname{det}\gamma_{ab}$
(4)

where $\gamma_{ab}$ is the unit $S^{2m}$ sphere metric222We are not assuming
that $\gamma_{ab}$ is the standard round sphere metric. with covariant
derivative $D_{a}$. All angular indices $(a,b,c\dots)$ are raised and lowered
with $\gamma_{ab}$.

We can now define an asymptotically flat spacetime by imposing the following
falloff conditions on the metric

$g_{uu}=\mathcal{O}(1),\quad\quad
g_{ur}=-1+\mathcal{O}\left(\frac{1}{r}\right),\quad\quad
g_{ua}=\mathcal{O}(1),\quad\quad g_{ab}=\mathcal{O}(r^{2})$ (5)

The asymptotic symmetry group is the group of all non-trivial diffeomorphisms
preserving the gauge-fixing conditions (4) and the asymptotic falloffs (5).
Such diffeomorphisms are generated by the vector

$\displaystyle\zeta^{u}$ $\displaystyle=f(z)+\frac{u}{2m}D_{a}Y^{a}(z)+\ldots$
(6) $\displaystyle\zeta^{a}$
$\displaystyle=-\frac{1}{r}D^{a}f(z)+Y^{a}(z)-\frac{u}{2mr}D^{a}D_{b}Y^{b}(z)+\ldots$
$\displaystyle\zeta^{r}$
$\displaystyle=\frac{1}{2m}D^{2}f(z)-\frac{r}{2m}D_{a}Y^{a}(z)+\frac{u}{4m^{2}}D^{2}D_{a}Y^{a}(z)+\ldots$

Transformations with $Y^{a}=0$ and an arbitrary function $f$ on the sphere are
known as supertranslations and they are a generalization of the global
translations. Transformations with $f=0$ and an arbitrary smooth vector
$Y^{a}$ on the sphere are known as Diff$(S^{2m})$ and they are a
generalization of the global Lorentz transformations. Note that not all
$\zeta$ are asymptotic killing vector fields

$\lim_{r\rightarrow\infty}\nabla_{(a}\zeta_{b)}\neq 0$ (7)

but they all are asymptotically divergence-free

$\lim_{r\rightarrow\infty}\nabla_{a}\zeta^{a}=0$ (8)

In $d=4$ an alternative and more restrictive description of asymptotic
flatness has been previously proposed, leading to the requirement that $Y$ is
a conformal killing vector (CKV) of the sphere Barnich:2010ojg ;
Barnich:2011mi ; Kapec:2014opa . In $d=4$, there are infinitely many local
solutions $Y$ to the CKV equation, corresponding to infinitely many
transformations generated by $\zeta$. Such transformations are called
superrotations and they generalize the Lorentz transformations given by global
solutions to the CKV equation. However the number of independent constraints,
imposed by the CKV equation, grows with the dimensions of the space. In $d=4$
the CKV equations are the Cauchy-Riemann equations and so there are infinite
independent local solutions. For $d>4$, the system of equations is over-
determined and there are only a finite number of independent solutions: the
global Lorentz transformations. In dimensions higher than four, since one
cannot extend Lorentz transformations to superrotations, the Diff$(S^{2m})$
seem to be the proper asymptotic symmetries to link to the subleading soft
graviton theorem. Therefore, here we will consider arbitrary smooth vectors
$Y_{a}$ of the sphere. The relationship between superrotations and
Diff$(S^{2})$ in $d=4$ has been studied in Donnay:2020guq  and it involves a
shadow transform. For $d>4$ such shadow transform is not well defined and this
is consistent with the fact that superrotations don’t exist in higher
dimensions.

##  3 From soft theorems to asymptotic symmetries

Let us start by considering the leading soft graviton theorem

$\lim_{\omega\rightarrow
0}\mathcal{M}_{n+n^{\prime}+1}\left(q=\omega\hat{q};p_{1},\ldots,p_{n+n^{\prime}}\right)=S^{(1)}\mathcal{M}_{n+n^{\prime}}\left(p_{1},\ldots,p_{n+n^{\prime}}\right)$
(9)

where

$S^{(1)}\equiv\frac{\kappa}{2}\left(\sum_{k=n+1}^{n+n^{\prime}}\frac{\varepsilon_{\mu\nu}p_{k}^{\mu}p_{k}^{\nu}}{p_{k}\cdot
q}-\sum_{k=1}^{n}\frac{\varepsilon_{\mu\nu}p_{k}^{\mu}p_{k}^{\nu}}{p_{k}\cdot
q}\right)\\\ $ (10)

Such theorem can be recast into a Ward identity (for the derivation see
Kapec:2015vwa )

$\langle out|Q^{+}\mathcal{S}-\mathcal{S}Q^{-}|in\rangle=0$ (11)

where the charges can be decomposed into a soft and a hard part

$Q^{\pm}=Q_{H}^{\pm}+Q_{S}^{\pm}$ (12)

The hard part comes from the soft factor $S^{(1)}$ on right-hand side of eq
(9) after some manipulation, while the soft part comes from the left-hand side
of the equation. Therefore, $Q_{H}$ generates asymptotic symmetries on the
external hard states, while $Q_{S}$ is responsible for the creation of a soft
graviton. Focusing for now on $Q^{+}$, the hard charge takes the form

$\left\langle z_{n+1},\ldots\right|Q_{H}^{+}=\left\langle
z_{n+1},\ldots\right|\sum_{k=n+1}^{n+n^{\prime}}E_{k}f\left(z_{k}\right)$ (13)

which represent the action of supertranslations on each outgoing particle. The
soft charge is given by

$Q_{S}^{+}\propto\int
d^{2m}z\sqrt{\gamma}f(z)\prod_{l=m+1}^{2m-1}\left(D^{2}-(2m-l)(l-1)\right)I^{(m-2)}D^{a}D^{b}g_{ab}^{(m-2)}$
(14)

where $g^{(n)}_{\mu\nu}$ is the $n$-component of the metric in the $1/r$
expansion and we defined the integral operator $I^{(m-2)}=\left(\int
du\right)^{(m-2)}$. 333In the four-dimensional case $(m=1)$, we use the
following prescriptions:
$\prod_{l=m+1}^{2m-1}\left(D^{2}-(l-1)(2m-l)\right)=1$ and
$I^{(m-2)}=\partial_{u}$. In Kapec:2015vwa ; Colferai:2020rte  the following
commutation relation has been proposed

$\left[\left.g_{uu}^{(2m-1)}(z)\right|_{\mathcal{I}_{-}^{+}},C\left(u^{\prime},z^{\prime}\right)\right]=\frac{8\pi
iG}{m}\frac{\delta^{2m}\left(z-z^{\prime}\right)}{\sqrt{\gamma}}$ (15)

where $g_{ab}^{(-1)}=\left(\frac{1}{m}\gamma_{ab}D^{2}-2D_{a}D_{b}\right)C$.
Thanks to this commutation relation, it is possible to show that the charges
$Q=Q_{S}+Q_{H}$ generate supertanslations. The eq. (15) has not been proven
yet but it is consistent with the four-dimensional case He:2014laa . In order
to derive the formula in higher dimensions, a renormalization will be
required.

Let us now consider the subleading soft graviton theorem

$\lim_{\omega\rightarrow
0}\left(1+\omega\partial_{\omega}\right)\mathcal{M}_{n+n^{\prime}+1}\left(q=\omega\hat{q};p_{1},\ldots,p_{n+n^{\prime}}\right)=S^{(2)}\mathcal{M}_{n+n^{\prime}}\left(p_{1},\ldots,p_{n+n^{\prime}}\right)$
(16)

where

$S^{(2)}=-\frac{i\kappa}{2}\left[\sum_{k=n+1}^{n+n^{\prime}}\frac{p_{k\mu}\varepsilon^{\mu\nu}q^{\lambda}J_{k\lambda\nu}}{p_{k}\cdot
q}-\sum_{k=1}^{n}\frac{p_{k\mu}\varepsilon^{\mu\nu}q^{\lambda}J_{k\lambda\nu}}{p_{k}\cdot
q}\right]$ (17)

The operator $\left(1+\omega\partial_{\omega}\right)$ projects out the pole
$\frac{1}{\omega}$ at the leading order in $S^{(1)}$.

The subleading soft theorem can also be recast as a Ward identity (for the
derivation see Colferai:2020rte ). The hard charge of such identity is given
by

$\displaystyle\left\langle
out\right|Q_{H}^{+}=i\sum_{k=n+1}^{n+n^{\prime}}\left(Y^{a}\left(z_{k}\right)\partial_{z^{a}_{k}}-\frac{E_{k}}{2m}D_{a}Y^{a}\left(z_{k}\right)\partial_{E_{k}}\right)\left\langle
out\right|$ (18)

which represent the action of a Diff$(S^{2m})$ on each outgoing particle. The
soft charge is given by

$\displaystyle Q^{+}_{S}\propto$ $\displaystyle\int du\int
d^{2m}z\sqrt{\gamma}\text{
}uD.Y\prod_{l=m+1}^{2m-1}\left(D^{2}-(l-1)(2m-l)\right)I^{(m-2)}\left(D^{a}D^{b}g_{ab}^{(m-2)}\right)$
(19)

Using again the commutation relation (15), we are able to show that such
charges generate Diff$(S^{2m})$.

##  4 Conclusions and outlook

We examined the relation between asymptotic symmetries and soft theorems at
leading and subleading order in higher-dimensional gravity. In order to
preserve the equivalence relation in $d>4$, the asymptotic charges need to be
renormalized. Such renormalization procedure has already been proven to be
effective in $d=4$ in Compere:2018ylh ; Compere:2020lrt ;
Chandrasekaran:2021vyu . The search for the right counterterms in $d>4$ is
currently an open problem that needs further investigation. However we already
know the final form of the renormalized charges since we can derive them from
soft theorems. Indeed, we can recast the soft graviton theorems as Ward
Identities. We then argued that such identities are associated to asymptotic
symmetries, provided a suitable commutation relation between metric fields
holds. Recent analyses also support the form of asymptotic charges we derived
Campoleoni:2020ejn . As a result, supertranslations and Diff($S^{2m}$) are
symmetries of gravitational scattering.

There are still many other open topics that need to be tackled in the future.
Soft theorems and asymptotic symmetries are part of a triangular equivalence
relation called IR Triangle. The last corner of the triangle is a memory
effect and it represents an exciting experimental prospect for the coming
future of the gravitational wave physics. The spin memory effect has been
studied in relation to the subleading soft graviton in $d=4$ in
Pasterski:2015tva . It would be interesting to examine such effect in higher
dimensions.

Moreover asymptotic symmetries and soft theorems play a central role in the
celestial holography program. Celestial holography proposes a duality between
gravitational scattering in asymptotically flat spacetimes and a conformal
field theory living in two dimensions lower on the celestial sphere (for
reviews of the topic see Pasterski:2021rjz ; Raclariu:2021zjz ;
McLoughlin:2022ljp ). This field of research has been extensively studied in
$d=4$ but it can also be generalized to higher dimensions Kapec:2017gsg ;
Pasterski:2017kqt  where further investigations are needed.

Lastly, the sub-subleading soft graviton theorem has been recently studied in
relation to asymptotic symmetries in $d=4$ Campiglia:2016efb ;
Campiglia:2016jdj ; Freidel:2021dfs . It would be worthwhile to also extend
such analyses to arbitrary dimensions.

## Acknowledgements

We thank C. CorianÃ for collaborating to related analysis. The work of S.L. is
partly supported by INFN Iniziativa Specifica QFT-HEP.

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