A Donsker-type Theorem for Log-likelihood Processes

TL;DR

This paper establishes a Donsker-type weak convergence result for the log-likelihood processes of a family of probability measures associated with a semimartingale, showing convergence to a Gaussian process under certain regularity conditions.

Contribution

It introduces a novel weak convergence theorem for log-likelihood processes in a semimartingale setting, extending classical results to more general stochastic processes.

Findings

Log-likelihood processes converge weakly to a Gaussian process

Convergence holds under conditions involving entropy and Hellinger processes

Results apply to a broad class of semimartingales

Abstract

Let $(\Omega, \mathcal{F}, (\mathcal{F})_{t\ge 0}, P)$ be a complete stochastic basis, $X$ a semimartingale with predictable compensator $(B, C, \nu)$. Consider a family of probability measures $\mathbf{P}=( {P}^{n, \psi}, \psi\in \Psi, n\ge 1)$, where $\Psi$ is an index set, $ {P}^{n, \psi}\stackrel {loc} \ll{P}$, and denote the likelihood ratio process by $Z_t^{n, \psi} =\frac{dP^{n, \psi}|_{\mathcal{F}_t}}{d P|_{\mathcal{F}_t}}$. Under some regularity conditions in terms of logarithm entropy and Hellinger processes, we prove that $\log Z_t^{n}$ converges weakly to a Gaussian process in $\ell^\infty(\Psi)$ as $n\rightarrow\infty$ for each fixed $t>0$.

Equations

$$C_t=⟨X^c,X^c{⟩}_t.$$

$$\mu (\omega ,dt,dx)=\sum _s{1}_{(\Delta X_s(\omega )\ne 0)}{\epsilon }_{(s,\Delta X_s(\omega ))}(dt,dx)$$

$$W\ast {\mu }_t(\omega )$$

$$\displaystyle=\left\{\begin{array}[]{ll}\int_{0}^{t}\int_{\mathbb{R}}W(\omega,s,x)\mu(\omega;ds,dx)&\mbox{if}\int_{0}^{t}\int_{\mathbb{R}}|W(\omega,s,x)|\mu(\omega;ds,dx)<\infty,\\ \\ +\infty&\mbox{otherwise}.\end{array}\right.$$

$$\displaystyle\langle W\ast(\mu-\nu)\rangle_{t}=\big{(}W-\hat{W}\big{)}^{2}\ast\nu_{t}+\sum_{s\leq t}(1-a_{s})\hat{W}^{2}_{s},$$

$$\hat{W}=\hat{W}(\omega ,t)={\int }_{\mathbb{R}}W(\omega ,t,x)\nu (\omega ,\{t\}\times dx)$$

$$a_t(\omega )=\nu (\omega ,\{t\}\times \mathbb{R}).$$

$$\displaystyle E\big{(}\mathbf{1}_{(\Delta X_{t}=0)}\big{|}\mathcal{F}_{t-}\big{)}=1-a_{t}.$$

$$P^{\prime }\stackrel{loc}{\ll }P,$$

$$Z_t=\frac{d{P}^{\prime }{|}_{{\mathcal{F}}_t}}{dP{|}_{{\mathcal{F}}_t}}.$$

$$\displaystyle Z_{t}=\left\{\begin{array}[]{ll}Z_{0}e^{N_{t}-\frac{1}{2}\beta^{2}\cdot C_{t}}\prod_{s\leq t}(1+\Delta N_{s})e^{-\Delta N_{s}},&\quad(\omega,t)\in\triangle,\\ 0,&\quad\mbox{otherwise}.\end{array}\right.$$

$$\displaystyle N_{t}=\beta\cdot X_{t}^{c}+\Big{(}Y-1+\frac{\hat{Y}-a}{1-a}\mathbf{1}_{(a<1)}\Big{)}\ast(\mu-\nu)_{t}$$

$$\sigma =\inf \{t:{\hat{Y}}_t>1\text{mbox}or{\hat{Y}}_t<1\text{mbox}and{a}_t=1\},$$

$$\displaystyle H_{t}=\frac{1}{2}\beta^{2}\cdot C_{t}+\big{(}1-\sqrt{Y}\big{)}^{2}\mathbf{1}_{[0,\sigma)}\ast\nu_{t}+\sum_{s\leq t}\Big{(}\sqrt{1-a_{s}}-\sqrt{1-\hat{Y}_{s}}\Big{)}^{2}\mathbf{1}_{(s<\sigma)},$$

$$T_n=\inf \{t:H_t\ge n\},$$

$$△=[0,\sigma )\cap ({\cup }_n[0,T_n]).$$

$$Y=M_{\mu}^{P}\Big{(}\frac{Z}{Z_{-}}|\tilde{\mathcal{P}}\Big{)}.$$

$$L_t=\log Z_t.$$

$$L_t$$

$$P^{n,\psi }\stackrel{\text{loc}}{\ll }P$$

$$P\ll Q,P^{\prime }\ll Q$$

$$Z=\frac{dP}{dQ},Z^{\prime }=\frac{d{P}^{\prime }}{dQ}$$

$$\displaystyle\rho^{2}(P,P^{\prime})=\frac{1}{2}\int_{\Omega}\big{(}\sqrt{Z}-\sqrt{Z^{\prime}}\big{)}^{2}dQ.$$

$${\rho }^2(P,P^{\prime })=1-E_Q\sqrt{Z{Z}^{\prime }}$$

$$P\stackrel{loc}{\ll }Q,P^{\prime }\stackrel{loc}{\ll }Q$$

$$Z_t=\frac{dP{|}_{{\mathcal{F}}_t}}{dQ{|}_{{\mathcal{F}}_t}},Z_t^{\prime }=\frac{d{P}^{\prime }{|}_{{\mathcal{F}}_t}}{dQ{|}_{{\mathcal{F}}_t}}.$$

$$h(\alpha ;P,P^{\prime }{)}_0=0$$

$$h(\alpha ;P,P^{\prime }{)}_t=1_{\cup [0,S_n]}\cdot h(\alpha ;P,P^{\prime }{)}_t$$

$$Y(\alpha {)}_t+Y(\alpha {)}_{-}\ast {\nu }_t\text{mbox}islocalmartingale$$

$$S_n=\inf \{t:Z_t>n\text{mbox}or{Z}_t^{\prime }>n\}$$

$$Y(\alpha {)}_t=Z_t^{\alpha }(Z_t^{\prime }{)}^{1-\alpha }.$$

Full text

1. Introduction and Preliminaries

The celebrated Donsker theorem is a functional extension of the central limit theorem in probability theory. Plenty of research on this topic has come out in the past decades. The reader is referred to classic books and papers like Dudley [3], Gine and Zinn [4], Ossiander [10], Andersen et. al [1], Liptser and Shiryaev [7], van der Geer [11], Billingsley [2], Jacod and Shiryaev [5] for both theoretical framework and wide applications. A primary purpose of the present paper is to establish a certain Donsker theorem for log-likelihood processes indexed by an arbitrary set. In this section, we first introduce some basic notions about log-likelihood processes and martingale representation property.

Throughout this paper, we follow the standard definitions and notations of martingale theory, which can be found in the book by Jacod and Shiryaev [5]. Let $(\Omega,\mathcal{F},(\mathcal{F})_{t\geq 0},P)$ be a complete stochastic basis. Fix a semimartingale $X$ on it, and assume that all $P$-martingales have a representation property relative to $X$. Denote by the triplet $(B,C,\nu)$ the predictable characteristic of $X$ (associated to some bounded truncation function). More precisely, if $\Delta X_{t}=X_{t}-X_{t-}$ denotes the jump of $X$ at time $t$, then $X_{t}-\sum_{s\leq t}(\triangle X_{s}-h_{\tau}(\triangle X_{s}))$ , where $h_{\tau}(x)=x1_{(|x|\leq\tau)}$, is a special semimartingale, which can be uniquely divided into a bounded variation process and a local martingale process. The $B$ is a bounded variation process of $X-\sum_{s\leq\cdot}(\triangle X_{s}-h_{\tau}(\triangle X_{s}))$. Let $X^{c}$ be the continuous local martingale part of $X$, then

[TABLE]

Let $\mu$ be the jump measure of $X$ defined by

[TABLE]

where $\varepsilon_{(s,\Delta X_{s}(\omega))}$ denotes the Dirac measure at point $(s,\Delta X_{s}(\omega))$. The $\nu$ is the unique predictable compensator of $\mu$ (up to a $P$-null set). Namely, $\nu$ is a predictable random measure such that for any predictable function111  Let $\tilde{\Omega}=\Omega\times\mathbb{R}_{+}\times\mathbb{R}$, $\tilde{\mathcal{P}}=\mathcal{P}\otimes\mathcal{B}$, where $\mathcal{B}$ is a Borel $\sigma$-field on $\mathbb{R}$ and $\mathcal{P}$ a $\sigma$-field generated by all left continuous adapted processes on $\Omega\times\mathbb{R}_{+}$. The predictable function is a $\tilde{\mathcal{P}}$-measurable function on $\tilde{\Omega}$. $W$, $W\ast(\mu-\nu)=W\ast\mu-W\ast\nu)$ is a local martingale, where the $W\ast\mu$ is defined by

[TABLE]

(See Section 2.1 of Chapter 2 in Jacod and Shiryaev [5] for more details). Note the predictable quadratic variation is given by

[TABLE]

where

[TABLE]

and

[TABLE]

It follows from Corollary 1.19 of Chapter 2 in Jacod and Shiryaev [5] that $a_{t}=0$ is equivalent to the fact $X$ is a quasi-left continuous process222  A quasi-left continuous process $X$ is a càdlág adapted process such that for any increasing stoping times $T_{n}$ with limit $T$, $\lim_{n\to\infty}X_{T_{n}}=X_{T}.$. Specially, for a process with independent increments, $a_{t}=0$ means this process has no fixed time of discontinuity 333  $t$ is called as the fixed time of discontinuity if $P(\Delta X_{t}\neq 0)>0$.. Thus we may and do choose a good version of both $\hat{W}$ and $a$ such that $\hat{W}$ is the predictable projection of $W(\omega,t,\Delta X_{t})\mathbf{1}_{(\Delta X_{t}\neq 0)}$ and $a_{t}\leq 1$. In particular,

[TABLE]

Now consider another probability measure $P^{\prime}$ such that

[TABLE]

which means that for any $t\geq 0$, $P^{\prime}|_{\mathcal{F}_{t}}\ll P|_{\mathcal{F}_{t}}$. Define the likelihood ratio process

[TABLE]

It follows from Chapter III in Jacod and Shiryaev [5] that $Z_{t}$ is a local martingale.

Since by assumption all $P$-martingales have a representation property relative to $X$, according to Theorem 5.19 of Chapter III in Jacod and Shiryayev [5], $Z_{t}$ has the following representation: there is a predictable process $\beta$ and a nonnegative predictable function $Y$ on $\tilde{\Omega}$ such that

[TABLE]

Here

[TABLE]

and $\triangle$ is a random set defined as follows

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Note that $\beta,Y$ and $\triangle$ depend on $P^{\prime}$. In fact, $Y$ can be explicitly represented as follows. Let $M_{\mu}^{P}$ be a measure on $(\tilde{\Omega},\tilde{\mathcal{P}})$ where $\tilde{\Omega}:=\Omega\times\mathbb{R}_{+}\times\mathbb{R}$, $\tilde{\mathcal{P}}:=\mathcal{P}\otimes\mathcal{B}$, such that $M_{\mu}^{P}(W)=E(W*\mu)_{\infty}$ for all measurable nonnegative functions $W$. Then $Y$ is the conditional expectation of $\frac{Z}{Z_{-}}$ with respect to $\tilde{\mathcal{P}}$ under $M_{\mu}^{P}$, namely

[TABLE]

Define the log-likelihood process $L_{t}$ by

[TABLE]

This process has been a well-studied object in the context of both stochastic processes and statistical inferences. Obviously,

[TABLE]

Assume we are given a family of probability measures $\mathbf{P}^{n}=\{P^{n,\psi}:\psi\in\Psi\}$ on $(\Omega,\mathcal{F})$, indexed by an arbitrary non-empty set $\Psi$, and assume

[TABLE]

for every $n>0$ and $\psi\in\Psi$. We shall be mainly interested in the sequence of likelihood ratio processes $Z_{t}^{n,\psi}$. The main purpose of the paper is to establish a certain Donsker theorem for log-likelihood processes $\log Z_{t}^{n}$ in $\ell^{\infty}(\Psi)$ as $n\rightarrow\infty$, where we denote by $\ell^{\infty}(\Psi)$ the space of bounded real-valued functions defined on $\Psi$.

It seems hard to develop directly an invariance principle for $\log Z^{n}$ due to complicated structure. To the best of our knowledge, there are only few works in this area, such as Le Cam [6], Vostrikova [13] and so on. The reader can find some interesting results in Nishiyama [8] and [9], where $\log Z^{n}$ is assumed to be very special continuous semimartingales and discrete time semimartingales respectively. It is a challenging problem to extend Nishiyama’s work to general setting. To attack such a problem, we shall combine stochastic calculus techniques and chaining arguments with the Kakutani-Hellinger distance for probability measures. In particular, we shall characterize the regularity of $\ell^{\infty}(\Psi)$-valued log-likelihood processes in terms of the Kakutani-Hellinger distance and the Hellinger processes.

The rest of the paper is organized as follows. We will first make some necessary assumptions and then state our main results in the Section 2. The proof of main results is given in Section 3, which consists of several lemmas and two propositions.

2. Main Results

To state our main results, we need some more notations and make some technical assumptions. Start with the Kakutani-Hellinger distance between two probability measures $P$ and $P^{\prime}$. Assume that $Q$ is a third probability measure on $(\Omega,\mathcal{F})$ such that

[TABLE]

Let

[TABLE]

and define the Kakutani-Hellinger distance by

[TABLE]

It is easy to check that $\rho(P,P^{\prime})$ is a metric in the space of probability measures and does not depend on the probability measure $Q$. Note

[TABLE]

For $0<\alpha<1$, call $\breve{H}(\alpha;P,P^{\prime})=E_{Q}(Z^{\alpha}(Z^{\prime})^{1-\alpha})$ the Hellinger integral of order $\alpha$. We remark that $\breve{H}(\alpha;P,P^{\prime})\rightarrow 1$ as $\alpha\rightarrow 0$ if $P^{\prime}\ll P$.

Proceed to introduce the Hellinger processes. Assume that

[TABLE]

and define

[TABLE]

Then for each $0<\alpha<1$, there is a unique predictable increasing process $h(\alpha;P,P^{\prime})$, called the Hellinger process of order $\alpha$, such that

(i)

[TABLE]

(ii)

[TABLE]

(iii)

[TABLE]

where

[TABLE]

and

[TABLE]

One can extend the above Hellinger process to order zero and even to a general function. Given a function $\psi:\mathbb{R}\mapsto\mathbb{R}$ such that

[TABLE]

is bounded with convention $\frac{0}{0}=0$ and $\psi(1)=0$. Denote

[TABLE]

then there is a predictable increasing process, denoted by $\imath(\psi;P,P^{\prime})$, such that

(i’)

[TABLE]

(ii’)

[TABLE]

(iii’)

[TABLE]

Call $\imath(\psi;P,P^{\prime})$ the Hellinger process of order 0 associated to $\psi$. In particular, if

[TABLE]

then we simply call $\imath(\psi;P,P^{\prime})$ the Hellinger process of order 0.

In general, it is rather complicated to compute $h(\alpha;P,P^{\prime})$. However, we fortunately have the following explicit formula in the special case $P^{\prime}\ll P$:

[TABLE]

In particular,

[TABLE]

[TABLE]

Our technical assumptions mainly involve three aspects: the predictable envelope of $\{Y^{n,\psi},\psi\in\Psi\}$, the Kakutani-Hellinger distance between probability measures $P^{n,\psi}$, and the size of index set $\Psi$.

For every $n>0$, denote the essence supremum $\overline{Y}^{n}(\Psi)=[\sup_{\psi\in\Psi}Y^{n,\psi}]_{\tilde{\mathcal{P}},M_{\nu}^{P}}$. This is the predictable envelope of $\{Y^{n,\psi},\psi\in\Psi\}$ used in Definitions 2.1 and 2.3 of Nishiyama [9].

Assumption 1**.**

For any $n>0$, $\psi\in\Psi$, $\triangle^{n,\psi}\equiv\Omega\times[0,1]$ and $0\leq a<1$. Moreover, $\{Y^{n,\psi},\psi\in\Psi\}$ attains their predictable envelope for every $n>0$, namely, there is a $\psi_{0}\in\Psi$ such that $Y^{n,\psi_{0}}=[\sup_{\psi\in\Psi}Y^{n,\psi}]_{\tilde{\mathcal{P}},M_{\nu}^{P}}$.

Assumption 2**.**

For every $\varepsilon>0$, as $n\rightarrow\infty$

[TABLE]

where $f_{1+\varepsilon}(x)=|x-1|1_{\{1/(1+\varepsilon)<x<1+\varepsilon\}^{c}}$.

There is a nonnegative definite continuous function $V_{t}$ on $\Psi\times\Psi$, such that as $n\rightarrow\infty$,

[TABLE]

and for every $\psi,\phi\in\Psi$,

[TABLE]

Let $\Psi$ be an arbitrary set, $\Delta_{\Pi}$ a positive rational number. $\Pi=\{\Pi(\varepsilon)\}_{\varepsilon\in(0,\Delta_{\Pi}]}$, is called a decreasing series of finite partitions (DFP) of $\Psi$ if

(i) each $\Pi(\varepsilon)=\{\Psi(\varepsilon;k):1\leq k\leq N_{\Pi}(\varepsilon)\}$ is a finite partition of $\Psi$, namely

[TABLE]

(ii) $N_{\Pi}(\Delta_{\Pi})=1$ and $\lim_{\varepsilon\downarrow 0}N_{\Pi}(\varepsilon)=\infty$;

(iii) $N_{\Pi}(\varepsilon)\geq N_{\Pi}(\varepsilon^{\prime})$ as $\varepsilon\leq\varepsilon^{\prime}$.

Given a $0<\varepsilon\leq\Delta_{\Pi}$, the $\varepsilon$-entropy $H_{\Pi}(\varepsilon)$ is defined by

[TABLE]

Assumption 3**.**

There exists a decreasing series of finite partitions, $\Pi$, of $\Psi$ such that as $n\rightarrow\infty$

[TABLE]

and

[TABLE]

where

[TABLE]

We are now ready to state our main result as follows.

Theorem 2.1**.**

Under Assumptions 1, 2 and 3, we have

[TABLE]

where $G$ stands for a Gaussian element in $\ell^{\infty}(\Psi)$, each $d$-dimensional marginal $(G^{\psi_{1}},\cdots,G^{\psi_{d}})$ is a normal random vector with mean

[TABLE]

and covariance structure

[TABLE]

The proof will be given in Section 2. For sake of comparison, we review an earlier result due to Nishiyama [9] in the discrete time case. Let $(\mathcal{F}_{i})_{i\geq 0}$ be a discrete time stochastic basis, and $\mathbf{P}^{n}=\{P^{n,\psi}:\psi\in\Psi\}$ a family of probability measures on $(\Omega,\mathcal{F})$, such that

[TABLE]

Define

[TABLE]

and

[TABLE]

Nishiyama [9] studied weak convergence for log-likelihood processes $\log W^{n}_{n}$ in $\ell^{\infty}(\Psi)$ and obtained a similar result to (2.13) under some integrability assumptions involving $\xi^{n}$’s and entropy conditions. More specifically, assume

(i) for every $\varepsilon>0$

[TABLE]

(ii) there exists a decreasing series of finite partitions, $\Pi$, of $\Psi$ such that

[TABLE]

and

[TABLE]

(iii) there is a $V:\Psi\times\Psi\rightarrow R$ such that

[TABLE]

and for $\psi,\phi\in\Psi$

[TABLE]

Then

[TABLE]

where $G$ stands for a Gaussian element in $l^{\infty}(\Psi)$, each $d$-dimensional marginal $(G^{\psi_{1}},\cdots,G^{\psi_{d}})$ is a normal random vector with mean

[TABLE]

and covariance structure

[TABLE]

To conclude the Introduction, two more remarks are given .

Remark 2.2*.*

Observe in the discrete time case the Hellinger process can be computed as follows.

[TABLE]

and

[TABLE]

Thus there is to some extent a similarity between our assumptions in Theorem 2.1 and Nishiyama’s assumptions. However, it seems neater to use the Hellinger processes in continuous time case.

The integrability condition (Assumption 3) of partitioning entropy plays an important role in the proof of Theorem 2.1. It is possible to use the metric entropy condition, but we need to introduce a suitable pseudo-metric in the index set $\Psi$. The Hellinger processes would also be very likely a good candidate.

Remark 2.3*.*

It is rather interesting to consider the limiting behavior of the process $\log Z^{n}$ in $\ell^{\infty}([0,1]\times\Psi)$. To this end, we need to establish a tightness criterion in the space $[0,1]\times\Psi$. This is more complicate, and will be left to the future work.

3. Proofs

Let us start with a decomposition. Observe that

[TABLE]

It is easy to see

[TABLE]

and so we have

[TABLE]

Let $\mu^{n,\psi}$ be the jump measure of $L^{n,\psi}$ defined by

[TABLE]

and $\nu^{n,\psi}$ the corresponding predictable compensator. Then for any predictable function $W(\omega,t,x)$,

[TABLE]

and so by the fact that $1-a_{s}$ is the predictable projection of $\mathbf{1}_{(\Delta X_{s}=0)}$,

[TABLE]

Given a positive number $\tau$, consider the truncation function

[TABLE]

and define

[TABLE]

Thus combined together, we easily have a canonical decomposition

[TABLE]

For simplicity of writing, let

[TABLE]

[TABLE]

[TABLE]

Thus we have

[TABLE]

The proof of Theorem 2.1 will consist of a series of lemmas and propositions.

Lemma 3.1**.**

Under Assumptions 1, 2 and 3, we have for each $\psi\in\Psi$ and $\tau>0$, as $n\rightarrow\infty$

[TABLE]

Proof.

Set

[TABLE]

Then

[TABLE]

Also, for any $\delta>0$

[TABLE]

By the Lenglart domination property (see page 35 of Jacod and Shiryaev [5]),

[TABLE]

Note for $\delta<1$ there is a positive constant $c_{\delta}$ such that for any $x>0$

[TABLE]

so

[TABLE]

On the other hand, for each $\varepsilon\in(0,1)$

[TABLE]

Again, by the Lenglart domination property, it follows for any $\eta>0$

[TABLE]

Letting $n\rightarrow\infty$ and then $\eta\rightarrow 0$, we have

[TABLE]

In combination, we have proved the desired statement. ∎

Lemma 3.2**.**

Under Assumptions 1, 2 and 3, we have for each $\psi\in\Psi$, as $n\rightarrow\infty$

[TABLE]

[TABLE]

[TABLE]

Consequently,

[TABLE]

Proof.

Obviously, for any $\varepsilon>0$

[TABLE]

Also, by Assumption 2

[TABLE]

The desired (3.8) holds.

Observe an elementary inequality: for any $0<\varepsilon<1$, there is a positive constant $c_{\varepsilon}$ such that

[TABLE]

Then it follows

[TABLE]

[TABLE]

[TABLE]

Thus under the Assumption 2, we have by letting $n\rightarrow\infty$ and then $\varepsilon\rightarrow 0$

[TABLE]

For (3.10), note

[TABLE]

where $\Upsilon^{n,\psi}_{s}$ is as in (3.6). Then it easily follows

[TABLE]

[TABLE]

[TABLE]

Thus under the Assumption 2, we have by first letting $n\rightarrow\infty$ and then $\varepsilon\rightarrow 0$

[TABLE]

The proof is now complete. ∎

Lemma 3.3**.**

Under Assumptions 1, 2 and 3, we have for each $\psi\in\Psi$ and $\tau>0$, as $n\rightarrow\infty$

[TABLE]

Proof.

First, observe the quadratic variation of $h_{\tau}\ast(\mu^{n,\psi}-\nu^{n,\psi})_{t}$ is given by

[TABLE]

We shall prove that $\langle h_{\tau}\ast(\mu^{n,\psi}-\nu^{n,\psi})\rangle_{t}$ converges in probability to zero below. Note there is a $\delta>0$ such that for any $\varepsilon<\delta$

[TABLE]

where $0<c_{\varepsilon}<\infty$. Thus it follows for any $\varepsilon<\delta$

[TABLE]

Hence letting $n\rightarrow\infty$ and then $\varepsilon\rightarrow 0$ immediately yields

[TABLE]

A similar argument shows

[TABLE]

Combined together, we have the desired statement. ∎

Lemma 3.4**.**

Under Assumptions 1, 2 and 3, we have for each $\psi\in\Psi$ and $\tau>0$, as $n\rightarrow\infty$

[TABLE]

Proof.

Note there is a $\delta>0$ such that for any $\varepsilon<\delta$

[TABLE]

where $0<c_{\varepsilon}<\infty$. Thus it follows for any $\varepsilon<\delta$

[TABLE]

This implies

[TABLE]

Since it was proved $2(1-\sqrt{Y^{n,\psi}})^{2}\ast\nu_{t}\stackrel{{\scriptstyle P}}{{\longrightarrow}}0$, then we have

[TABLE]

Similarly, we have

[TABLE]

Combined together, the proof is complete. ∎

Proposition 3.5**.**

Under Assumptions 1, 2 and 3, every finite-dimensional marginal of $L^{n}$ converges weakly.

Proof.

From Lemmas 3.1-3.4,

[TABLE]

has non-degenerate limiting finite-dimensional marginal laws, and the other part of $L^{n}$ asymptotically vanish.

For every $\psi$, the process

[TABLE]

is a continuous semimartingale. Its predictable characterstics $(\hat{B}^{\psi},\hat{C}^{\psi},0)$ are

[TABLE]

[TABLE]

By Lemma 3.2 and Assumption 2, there is a non-decreasing continuous function $V$, such that $V_{0}=0$,

[TABLE]

[TABLE]

for every $\psi,\phi\in\Psi$.

The proposition is now concluded by Theorem VIII.3.6 of Jacod and Shiryaev [5]. ∎

Next we turn to verifying uniform tightness.

Lemma 3.6**.**

Under Assumptions 1, 2 and 3, we have for each $\tau>0$, as $n\rightarrow\infty$

[TABLE]

Proof.

Recall

[TABLE]

Let us prove

[TABLE]

and

[TABLE]

For (3.19), note

[TABLE]

Thus we need only to prove

[TABLE]

and

[TABLE]

Let us first look at (3.21). Set

[TABLE]

Then

[TABLE]

For any $\varepsilon,\eta>0$,

[TABLE]

Note $x-1>\log x$ if $\log x>0$. Then

[TABLE]

Recall the definition of $\imath(f_{1+\varepsilon},P,P^{n,\psi})$ and Assumptions 1 and 2, we can obtain (3.21). The proofs of $(\ref{Eq4})$ and $(\ref{Eq2})$ are similar. ∎

Lemma 3.7**.**

Under Assumptions 1, 2 and 3, for any $\varepsilon,\eta>0$, there is a $\delta>0$ and a partition $\Pi(\delta)=\{\Psi(\delta),\,1\leq k\leq N(\delta)\}$ such that

[TABLE]

Proof.

Let us fix $\varepsilon,\eta>0$. First, note

[TABLE]

Let

[TABLE]

and

[TABLE]

We shall treat $J^{n,1,\psi}_{t}$ and $J^{n,2,\psi}_{t}$ separately below. Let us only focus on the $J^{n,2,\psi}_{t}$ since the $J^{n,1,\psi}_{t}$ is similar and simpler.

According to Assumption 3, there is a sufficiently large positive finite constant $K$ such that

[TABLE]

Thus we only need to condition on the event $\{\mathfrak{H}^{n}\|_{\Pi,t}>K\}$. In particular, we shall prove

[TABLE]

Assuming (3.28), we can take $\delta$ so small that

[TABLE]

from which it follows by the Markov inequality

[TABLE]

This in turn together with (3.27) implies

[TABLE]

It remains to prove (3.28). For every integer $p\geq 0$, construct a nested refinement partition $\Pi(2^{-p}\delta)=\{\Psi(2^{-p}\delta;k),1\leq k\leq N_{\Pi}(2^{-p}\delta)\}$ of $\Psi$, and then choose an element $\psi_{p,k}$ from each partitioning set $\Psi(2^{-p}\delta;k)$ in such a way that

[TABLE]

For every $\psi\in\Psi$ and each $p\geq 0$, define $\pi_{p}\psi=\psi_{p,k}$ and $\Pi_{p}\psi=\Psi(2^{-p}\delta;k)$ whenever $\psi\in\Psi(2^{-p}\delta;k)$. Obviously, $\Pi_{p}\psi\subseteq\Pi_{p-1}\psi$. Define

[TABLE]

Note $W(\Pi_{p}\psi)\leq W(\Pi_{p-1}\psi)$. Set

[TABLE]

and

[TABLE]

and for $p\geq 1$

[TABLE]

[TABLE]

It is easy to see

[TABLE]

and for each $p\geq 1$

[TABLE]

Hence it follows for any $q\geq 1$

[TABLE]

Note $\pi_{0}\psi=\pi_{0}\phi$ if $\psi,\phi\in\Psi(\delta,k)$, and so we have

[TABLE]

It is now enough to show

[TABLE]

We have the following identity

[TABLE]

Denote

[TABLE]

We shall only establish (3.32) for $M^{n,p,\psi}$ since the other three terms in RHS of (3) can be similarly treated.

Obviously, $M^{n,p,\psi}$ is a local martingale, and

[TABLE]

On the other hand, the predictable quadratic variation of $M^{n,p,\psi}_{t}$ satisfies

[TABLE]

By an elementary calculation, for any $\gamma\in(0,1)$ there is a constant $c_{\gamma}$ such that

[TABLE]

whenever $x,y\in[1-\gamma,1+\gamma]$. Then it follows

[TABLE]

By Bernstein-Freedman’s inequality (see Lemma 3.2 of Nishiyama [8]) for local martingale with bounded jumps, it follows for $\varepsilon>0$,

[TABLE]

This, in turn together with Lemma 2.2.10 of van der Vaart and Wellner [12], yields

[TABLE]

and

[TABLE]

Thus (3.28) is obtained, and so complete the proof. ∎

Lemma 3.8**.**

Under Assumptions 1, 2 and 3, for any $\varepsilon,\eta>0$, there is a $\delta>0$ and a partition $\Pi(\delta)=\{\Psi(\delta),\,1\leq k\leq N(\delta)\}$ such that

[TABLE]

Proof.

Recall

[TABLE]

It is enough to prove the following two statements

[TABLE]

and

[TABLE]

We shall concentrate on proving (3.34) below since (3.35) is similar. The proof is completely similar to that of Lemma 3.7 with some minor modifications. For every integer $p\geq 0$, choose an element $\psi_{p,k}$ from each partitioning set $\Psi(2^{-p}\delta;k)$ in such a way that

[TABLE]

and define $\pi_{p}\psi=\psi_{p,k}$ and $\Pi_{p}\psi=\Psi(2^{-p}\delta;k)$ whenever $\psi\in\Psi(2^{-p}\delta;k)$. Note

[TABLE]

then a main step is to prove

[TABLE]

To this end, for $p\geq 0$, set

[TABLE]

Obviously, $\Gamma(\Pi_{p}\psi)\leq\Gamma(\Pi_{p-1}\psi)$. Define

[TABLE]

[TABLE]

[TABLE]

where $\alpha_{p}$ is as in (3.31).

Note we have the following identity

[TABLE]

and

[TABLE]

In addition, it is easy to see

[TABLE]

by Schwarz’s inequality. Thus

[TABLE]

Thus, (3.34) is proved. We complete the proof of this lemma. ∎

We can obtain the following proposition by Lemmas 3.6 - 3.8.

Proposition 3.9**.**

Under Assumptions 1, 2 and 3, for any $\varepsilon,\eta>0$, there is a $\delta>0$ and a partition $\Pi(\delta)=\{\Psi(\delta),\,1\leq k\leq N(\delta)\}$ such that

[TABLE]

The proof of Theorem 2.1. Proposition 3.9 implies the asymptotic equicontinuity of $L^{n}$, and the asymptotic marginal distribution of $L^{n}$ is obtained by Proposition 3.5. Then we can obtain Theorem 2.1 by these two propositions and Theorem 1.1 in Nishiyama [9].

Acknowledgments

The authors would like to thank the anonymous referees and the Associate Editor for careful reading and constructive comments. This work was supported by the National Natural Science Foundation of China (No.11371317, 11701331, 11731012, 11871425) , Fundamental Research Funds for Central Universities, Shandong Provincial Natural Science Foundation (No. ZR2017QA007) and Young Scholars Program of Shandong University.