Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process

Abstract

Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

1. Introduction

A Galton–Watson process can be described as follows:

\begin{equation*} Z_0=1,\quad Z_{n+1}= \sum_{i=1}^{Z_n} X_{n,i} \quad \text{for $ n \geq 0$,} \end{equation*}

where $X_{n,i}$ is the offspring number of the ith individual of the generation n. Moreover, the random variables $ (X_{n,i})_{i\geq 1} $ are independent of each other with common distribution law

\begin{equation*} \mathbb{P}(X_{n,i} =k ) = p_k,\quad k \in \mathbb{N},\end{equation*}

and are also independent of $Z_n$ .

An important task in statistical inference for Galton–Watson processes is to estimate the average offspring number of an individual m, usually termed the offspring mean. Clearly we have

\[ m=\mathbb{E}Z_1=\mathbb{E} X_{n,i} =\sum_{k=0}^\infty k p_k. \]

Let v denote the standard variance of $Z_1$ , that is,

\begin{equation*} v^2=\mathbb{E} (Z_1-m)^2.\end{equation*}

To avoid triviality, assume that $v> 0$ . For estimation of the offspring mean m, the Lotka–Nagaev [12, 14] estimator $Z_{n+1}/Z_{n}$ plays an important role. Throughout the paper we assume that

\[ p_0=0. \]

Then the Lotka–Nagaev estimator is well-defined $\mathbb{P}$ -a.s. For the Galton–Watson processes, Athreya [1] has established large deviations for the normalized Lotka–Nagaev estimator (see also Chu [3] for self-normalized large deviations); Ney and Vidyashankar [15, 16] and He [9] obtained sharp rate estimates for the large deviation behavior of the Lotka–Nagaev estimator; Maaouia and Touati [13] established a self-normalized central limit theorem (CLT) for the maximum likelihood estimator of m; Bercu and Touati [2] proved an exponential inequality for the Lotka–Nagaev estimator via self-normalized martingale methods. Alternative approaches for obtaining self-normalized exponential inequalities can be found in de la Peña, Lai, and Shao [4]. Despite the fact that the Lotka–Nagaev estimator is well studied, there is no result for self-normalized Cramér moderate deviations for the Lotka–Nagaev estimator. The main purpose of this paper is to fill this gap.

Let us briefly introduce our main result. Assume that $n_0, n \in \mathbb{N}$ . Notice that, by the classical CLT for independent and identically distributed (i.i.d.) random variables,

\[\textbf{X}_{n_n, n }=\biggl( \sqrt{Z_{n_0}} \biggl( \dfrac{Z_{n_0+1}}{Z_{ n_0}} -m \biggr),\ldots , \sqrt{Z_{ n_0+n-1}} \biggl( \dfrac{Z_{ n_0+n}}{Z_{n_0+n-1}} -m \biggr) \biggr)\]

asymptotically behaves like a vector of i.i.d. Gaussian random variables with mean 0 and variance $v^2$ (even if $n_0$ depends on n) and the convergence rate to Gaussian distribution is exponential; see Kuelbs and Vidyashankar [11]. Because

\[\dfrac1n \sum_{k=n_0}^{n_0+n-1} Z_{ k} \biggl( \dfrac{Z_{ k+1}}{Z_{ k}} - m \biggr)^2\]

is an estimator of the offspring variance $v^2$ , it is natural to compare the self-normalized sum

\[M_{n_0,n}\,{:\!=}\, \dfrac{ {(n v^2)^{-1/2}} \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} ( {{Z_{ k+1}}/{Z_{ k}}} -m )}{\sqrt{ {(n v^2)^{-1}} \sum_{k=n_0}^{n_0+n-1} Z_{ k} ( {{Z_{ k+1}}/{Z_{ k}}} - m )^2 } }=\dfrac{ \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} ( {{Z_{ k+1}}/{Z_{ k}}} -m )}{\sqrt{\sum_{k=n_0}^{n_0+n-1} Z_{ k} ( {{Z_{ k+1}}/{Z_{ k}}} - m )^2 } }\]

to the tail of the Gaussian distribution. This is the main purpose of the paper. Assume that $ \mathbb{E} Z_1 ^{2+\rho}< \infty$ for some $ \rho \in (0, 1]$ . We prove the following self-normalized Cramér moderate deviations for the Lotka–Nagaev estimator. It holds that

(1) \begin{equation} \mathbb{P} (\!\pm \, M_{n_0,n}\geq x ) = (1-\Phi(x))(1+{\textrm{o}}(1) )\end{equation}

uniformly for $x \in [0, \, {\textrm{o}}( n^{\rho/(4+2\rho)} ))$ as $n\rightarrow \infty$ ; see Theorem 2.1. This type of result is user-friendly in the statistical inference of m, since in practice we usually do not know the variance $v^2$ or the distribution of $Z_1$ . Let $\kappa_n \in (0, 1)$ . Assume that

\begin{equation*} |\! \ln \kappa_n | ={\textrm{o}}\bigl( n^{\rho/(2+\rho)} \bigr), \quad n\rightarrow \infty .\end{equation*}

From (1) we can easily obtain a $1-\kappa_n$ confidence interval for m, for n large enough. Clearly, the right-hand side of (1) and $M_{n_0,n}$ do not depend on $v^2$ , so the confidence interval of m does not depend on $v^2$ ; see Proposition 3.1. Due to these significant advantages, the limit theory for self-normalized processes is attracting more and more attention. We refer to Jing, Shao, and Wang [10] and Fan et al. [8] for closely related results.

The paper is organized as follows. In Section 2 we present Cramér moderate deviations for the self-normalized Lotka–Nagaev estimator, provided that $(Z_n)_{n\geq0}$ can be observed. In Section 3 we present some applications of our results in statistics. The remaining sections are devoted to the proofs of theorems.

2. Main results

Assume that the total populations $(Z_{k})_{k\geq 0}$ of all generations can be observed. For $n_0, n \in \mathbb{N}$ , recall the definition of $M_{n_0,n}$ :

\begin{equation*}M_{n_0,n}= \dfrac{ \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} ( {{Z_{ k+1}}/{Z_{ k}}} -m )}{\sqrt{\sum_{k=n_0}^{n_0+n-1} Z_{ k} ( {{Z_{ k+1}}/{Z_{ k}}} - m )^2 } }.\end{equation*}

Here $n_0$ may depend on n. For instance, we can take $n_0$ as a function of n. We may take $n_0=0$ . However, in real-world applications it may happen that we know historical data $(Z_{k})_{ n_0 \leq k \leq n_0+n}$ for some $n_0\geq2$ , as well as the increment n of generation numbers, but do not know the data $(Z_{k})_{ 0 \leq k \leq n_0-1}$ . In such a case $M_{0,n}$ is no longer applicable to estimating m, whereas $M_{n_0,n}$ is suitable. Motivated by this problem, it would be better to consider the more general case $n_0\geq0$ instead of taking $n_0=0$ . As $(Z_k)_{k=n_0,\ldots,n_0+n}$ can be observed, $M_{n_0,n}$ can be regarded as a time-type self-normalized process for the Lotka–Nagaev estimator $Z_{k+1}/Z_{k}$ . The following theorem gives a self-normalized Cramér moderate deviation result for the Galton–Watson processes.

Theorem 2.1. Assume that $ \mathbb{E} Z_1 ^{2+\rho}< \infty$ for some $ \rho \in (0, 1]$ .

1. (i) If $\rho \in (0, 1)$ , then for all $x \in [0,\, {\textrm{o}}( \sqrt{n} ))$ ,

   (2) \begin{equation}\biggl|\ln\dfrac{\mathbb{P}(M_{n_0,n} \geq x)}{1-\Phi(x)} \biggr| \leq C_\rho \biggl( \dfrac{ x^{2+\rho} }{n^{\rho/2}} + \dfrac{ (1+x)^{1-\rho(2+\rho)/4} }{n^{\rho(2-\rho)/8}} \biggr) ,\end{equation}
   where $C_\rho$ depends only on the constants $\rho, v$ and $ \mathbb{E} Z_1 ^{2+\rho}$ .

2. (ii) If $\rho =1$ , then for all $x \in [0,\, {\textrm{o}}( \sqrt{n} ))$ ,

   (3) \begin{equation}\biggl|\ln\dfrac{\mathbb{P}(M_{n_0,n} \geq x)}{1-\Phi(x)} \biggr| \leq C \biggl( \dfrac{ x^{3} }{\sqrt{n} } + \dfrac{ \ln n }{\sqrt{n} } + \dfrac{ (1+x)^{1/4} }{n^{1/8}} \biggr) ,\end{equation}
   where C depends only on the constants v and $ \mathbb{E} Z_1 ^{3}$ .

In particular, inequalities (2) and (3) together imply that

(4) \begin{equation}\dfrac{\mathbb{P}(M_{n_0,n} \geq x)}{1-\Phi(x)} =1+{\textrm{o}}(1)\end{equation}

uniformly for $ x \in [0, \, {\textrm{o}}(n^{\rho/(4+2\rho)}))$ as $n\rightarrow \infty$ . Moreover, the same inequalities remain valid when

\[ \frac{\mathbb{P}( M_{n_0,n} \geq x)}{1-\Phi(x)} \]

is replaced by

\[ \frac{\mathbb{P}(M_{n_0,n} \leq -x)}{ \Phi (\!-x)}. \]

Notice that the mean of a standard normal random variable is 0. By the maximum likelihood method, it is natural to let $M_{n_0,n}=0$ ; then we have

\[\sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} \biggl( \frac{Z_{ k+1}}{Z_{ k}} -m \biggr)=0,\]

which implies that

\[\overline{m}_n \,{:\!=}\, \dfrac{1}{\sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} } \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} \biggl( \dfrac{Z_{ k+1}}{Z_{ k}} \biggr)\]

can be regarded as a random weighted Lotka–Nagaev estimator for m.

Equality (4) implies that $\mathbb{P}(M_{n_0,n} \leq x) \rightarrow \Phi(x)$ as n tends to $\infty$ . Thus Theorem 2.1 implies the central limit theory for $M_{n_0,n}$ . Moreover, equality (4) states that the relative error of normal approximation for $M_{n_0,n}$ tends to zero uniformly for $ x \in [0, \, {\textrm{o}}(n^{\rho/(4+2\rho)})) $ as $n\rightarrow \infty$ .

Theorem 2.1 implies the following moderate deviation principle (MDP) result for the time-type self-normalized Lotka–Nagaev estimator.

Corollary 2.1. Assume the conditions of Theorem 2.1. Let $(a_n)_{n\geq1}$ be any sequence of real numbers satisfying $a_n \rightarrow \infty$ and $a_n/ \sqrt{n} \rightarrow 0$ as $n\rightarrow \infty$ . Then, for each Borel set B,

\begin{equation*}- \inf_{x \in B^o}\dfrac{x^2}{2} \leq \liminf_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl( \dfrac{M_{n_0,n} }{ a_n } \in B \biggr) \leq \limsup_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl(\dfrac{ M_{n_0,n} }{ a_n } \in B \biggr) \leq - \inf_{x \in \overline{B}}\dfrac{x^2}{2} , \end{equation*}

where $B^o$ and $\overline{B}$ denote the interior and the closure of B, respectively.

Remark 2.1. From (2) and (3), it is easy to derive the following Berry–Esseen bound for the self-normalized Lotka–Nagaev estimator:

\begin{equation*}\sup_{x \in \mathbb{R} }|\mathbb{P}(M_{n_0,n} \leq x) - \Phi(x) | \leq \dfrac{ C_\rho }{n^{\rho(2-\rho)/8}},\end{equation*}

where $C_\rho$ depends only on the constants $\rho, v$ and $ \mathbb{E} Z_1 ^{2+\rho}$ . When $\rho> 1$ , by the self-normalized Berry–Esseen bound for martingales in Fan and Shao [6], we can get a Berry–Esseen bound of order $n^{- \rho/(6+2\rho) }$ .

The last remark gives a self-normalized Berry–Esseen bound for the Lotka–Nagaev estimator, while the next theorem presents a normalized Berry–Esseen bound for the Lotka–Nagaev estimator. Denote

\[H_{n_0, n}= \dfrac{1}{ \sqrt{n } v }\sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} \biggl( \dfrac{Z_{k+1}}{Z_{k}} -m \biggr).\]

Notice that the random variables $(X_{k,i})_{1\leq i\leq Z_k}$ have the same distribution as $Z_1$ , and that $(X_{k,i})_{1\leq i\leq Z_k}$ are independent of $Z_k$ . Then, for the Galton–Watson processes, it holds that

\[\mathbb{E} [ (Z_{k+1} -m Z_{k})^2 \mid Z_k] = \mathbb{E} \Biggl[ \Biggl( \sum_{i=1}^{Z_k} (X_{k,i} -m) \Biggr)^2 \biggm| Z_k \Biggr] = Z_{k} v^2.\]

It is easy to see that

\[H_{n_0, n}= \sum_{k=n_0}^{n_0+n-1} \frac{1}{ \sqrt{\, n v^2 / Z_{ k} }} \biggl( \frac{Z_{k+1}}{Z_{k}} -m \biggr) .\]

Thus $H_{n_0,n}$ can be regarded as a normalized process for the Lotka–Nagaev estimator $Z_{k+1}/Z_{k}$ . We have the following normalized Berry–Esseen bounds for the Galton–Watson processes.

Theorem 2.2. Assume the conditions of Theorem 2.1.

1. (i) If $\rho \in (0, 1)$ , then

   (5) \begin{equation} \sup_{x \in \mathbb{R}}|\mathbb{P}( H_{n_0,n} \leq x) - \Phi(x) | \leq \dfrac{ C_\rho }{ n^{\rho/2}},\end{equation}
   where $C_\rho$ depends only on $\rho, v$ and $ \mathbb{E} Z_1 ^{2+\rho}$ .

2. (ii) If $\rho =1$ , then

   (6) \begin{equation}\sup_{x \in \mathbb{R}}|\mathbb{P}( H_{n_0,n} \leq x) - \Phi(x) | \leq C \dfrac{ \ln n }{ \sqrt{n} },\end{equation}
   where C depends only on v and $ \mathbb{E} Z_1 ^{3}$ .

Moreover, the same inequalities remain valid when $H_{n_0,n}$ is replaced by $-H_{n_0,n}$ .

The convergence rates of (5) and (6) are identical to the best possible convergence rates of the Berry–Esseen bounds for martingales; see Theorem 2.1 of Fan [5] and the associated comment. Notice that $H_{n_0,n}$ is a martingale with respect to the natural filtration.

3. Applications

Cramér moderate deviations certainly have many applications in statistics.

3.1. p-value for hypothesis testing

Self-normalized Cramér moderate deviations can be applied to hypothesis testing of m for the Galton–Watson processes. When $(Z_{k})_{k=n_0,\ldots,n_0+n}$ can be observed, we can use Theorem 2.1 to estimate the p-value. Assume that $ \mathbb{E} Z_1 ^{2+\rho}< \infty$ for some $0 < \rho \leq 1$ , and that $m> 1$ . Let $(z_{k})_{k=n_0,\ldots,n_0+n}$ be the observed value of the $(Z_{k})_{k=n_0,\ldots,n_0+n}$ . In order to estimate the offspring mean m, we can make use of the Harris estimator [2] given by

\[\widehat{m}_n=\dfrac{\sum_{k=n_0}^{n_0+n-1} Z_{ k+1}}{\sum_{k=n_0}^{n_0+n-1}Z_{ k}}.\]

Then the observation for the Harris estimator is

\[\widehat{m}_n=\dfrac{\sum_{k=n_0}^{n_0+n-1} z_{ k+1}}{\sum_{k=n_0}^{n_0+n-1}z_{ k}}.\]

By Theorem 2.1, it is easy to see that

(7) \begin{equation} \dfrac{\mathbb{P}( M_{n_0,n} \geq x)}{1-\Phi(x)}=1+{\textrm{o}}(1)\quad \textrm{and} \quad \dfrac{\mathbb{P}(M_{n_0,n} \leq- x)}{1-\Phi(x)}=1+{\textrm{o}}(1)\end{equation}

uniformly for $ x \in [0, {\textrm{o}}( n^{\rho/(4+2\rho)} ))$ . Notice that $ 1-\Phi(x) = \Phi (\!-x). $ Thus, when $|\widetilde{m}_n|={\textrm{o}}( n^{\rho/(4+2\rho)} )$ , by (7), the probability $\mathbb{P}(M_{n_0,n} > |\widetilde{m}_n|)$ is almost equal to $2 \Phi (\!-|\widetilde{m}_n|) $ , where

\[ \widetilde{m}_n= \dfrac{ \sum_{k=n_0}^{n_0+n-1} \sqrt{z_{ k}} ( z_{ k+1}/z_{ k} -\widehat{m}_n )}{\sqrt{\sum_{k=n_0}^{n_0+n-1} z_{ k} ( z_{ k+1}/z_{ k} - \widehat{m}_n )^2 } }. \]

3.2. Construction of confidence intervals

Assume the data $(Z_{k})_{k\geq 0}$ can be observed. Cramér moderate deviations can also be applied to the construction of confidence intervals of m. We use Theorem 2.1 to construct confidence intervals.

Proposition 3.1. Assume that $\mathbb{E} Z_1 ^{2+\rho}< \infty$ for some $ \rho \in (0, 1]$ . Let $\kappa_n \in (0, 1)$ . Assume that

(8) \begin{equation} | \!\ln \kappa_n | ={\textrm{o}} \bigl( n^{\rho/(2+\rho)} \bigr) .\end{equation}

Let

\begin{align*}a_{n_0, n}&= \Biggl( \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{k}} \Biggr)^2- (\Phi^{-1}(1-\kappa_n /2) )^2 \sum_{k=n_0}^{n_0+n-1} Z_{k}, \\b_{n_0, n}&= 2 (\Phi^{-1}(1-\kappa_n /2) )^2 \sum_{k=n_0}^{n_0+n-1} Z_{k+1} - 2 \Biggl( \sum_{k=n_0}^{n_0+n-1} \dfrac{Z_{k+1}}{\sqrt{Z_{k }}} \Biggr)\Biggl( \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{k}} \Biggr), \\c_{n_0, n}&= \Biggl( \sum_{k=n_0}^{n_0+n-1} \dfrac{Z_{k+1}}{\sqrt{Z_{k }}} \Biggr)^2 - (\Phi^{-1}(1-\kappa_n /2) )^2 \sum_{k=n_0}^{n_0+n-1} \dfrac{Z_{k+1}^2}{Z_{k}}.\end{align*}

Then $[A_{n_0, n},B_{n_0, n}]$ , with

\begin{equation*}A_{n_0, n}=\dfrac{- b_{n_0, n} - \sqrt{b_{n_0, n}^2-4 a_{n_0, n}c_{n_0, n} }}{ 2 a_{n_0, n} }\end{equation*}

and

\begin{equation*}B_{n_0, n}=\dfrac{ -b_{n_0, n} + \sqrt{b_{n_0, n}^2-4 a_{n_0, n}c_{n_0, n} }}{ 2 a_{n_0, n} },\end{equation*}

is a $1-\kappa_n$ confidence interval for m, for n large enough.

Proof. Notice that $ 1-\Phi(x) = \Phi (\!-x). $ Theorem 2.1 implies that

(9) \begin{equation} \dfrac{\mathbb{P}( M_{n_0,n} \geq x)}{1-\Phi(x)}=1+{\textrm{o}}(1)\quad \textrm{and} \quad \dfrac{\mathbb{P}(M_{n_0,n} \leq- x)}{1-\Phi(x)}=1+{\textrm{o}}(1)\end{equation}

uniformly for $0\leq x={\textrm{o}}( n^ {\rho/(4+2\rho)} )$ ; see (4). Notice that the inverse function $\Phi^{-1}$ of a standard normal distribution function $\Phi$ has the following asymptotic expansion:

\[\Phi^{-1}(1-p)=\sqrt{\ln(1/p^2)-\ln\ln(1/p^2) -\ln(2\pi) }+ {\textrm{o}}(p) ,\quad p \searrow 0.\]

In particular, this says that for any positive sequence $(\kappa_n)_{n\geq1} $ that converges to zero, as $n\rightarrow \infty$ , we have

\[\Phi^{-1}( 1-\kappa_n/2) = \sqrt{2|\!\ln \kappa_n |}+{\textrm{o}}(\sqrt{ |\!\ln \kappa_n |}).\]

Thus, when $\kappa_n$ satisfies the condition (8), the upper $(\kappa_n/2)$ th quantile of a standard normal distribution is of order ${\textrm{o}}( n^ {\rho/(4+2\rho)} )$ . Then, applying (9) to the last equality, we complete the proof of Proposition 3.1. Notice that $A_{n_0, n}$ and $B_{n_0, n}$ are solutions of the following equation:

\[\dfrac{ \sum_{k=n_0}^{n_0+n-1} \sqrt{Z_{ k}} ( Z_{ k+1}/Z_{ k} -x )}{\sqrt{\sum_{k=n_0}^{n_0+n-1} Z_{ k} ( Z_{ k+1}/Z_{ k} - x )^2 } }=\Phi^{-1}(1-\kappa_n /2).\]

This completes the proof of Proposition 3.1.

Assume the parameter $v^2$ is known. When $v^2$ is known, we can apply normalized Berry–Esseen bounds (see Theorem 2.2) to construct confidence intervals.

Proposition 3.2. Assume that $\mathbb{E} Z_1 ^{2+\rho}< \infty$ for some $ \rho \in (0, 1]$ . Let $\kappa_n \in (0, 1)$ . Assume that

(10) \begin{equation} |\! \ln \kappa_n | ={\textrm{o}}(\! \ln n ) .\end{equation}

Then $[A_n,B_n]$ , with

\begin{equation*}A_n=\dfrac{ \sum_{k=n_0}^{n_0+n } Z_{k+1}/\sqrt{Z_k}-\sqrt{n } v \Phi^{-1}(1-\kappa_n/2) }{ \sum_{k=n_0}^{n_0+n } \sqrt{Z_{k}}}\end{equation*}

and

\begin{equation*}B_n=\dfrac{ \sum_{k=n_0}^{n_0+n } Z_{k+1}/\sqrt{Z_k}+\sqrt{n } v \Phi^{-1}(1-\kappa_n/2) }{ \sum_{k=n_0}^{n_0+n } \sqrt{Z_{k}}} ,\end{equation*}

is a $1-\kappa_n$ confidence interval for m, for n large enough.

Proof. Theorem 2.2 implies that

(11) \begin{equation} \dfrac{\mathbb{P}(H_{n_0,n} \geq x)}{1-\Phi(x)}=1+{\textrm{o}}(1)\quad \textrm{and} \quad \dfrac{\mathbb{P}(H_{n_0,n} \leq -x)}{1-\Phi(x)}=1+{\textrm{o}}(1)\end{equation}

uniformly for $0\leq x={\textrm{o}}( \sqrt{\ln n} )$ . The upper $(\kappa_n/2)$ th quantile of a standard normal distribution satisfies

\[\Phi^{-1}( 1-\kappa_n/2) = {\textrm{O}}(\sqrt{|\! \ln \kappa_n |} ),\]

which, by (10), is of order $ {\textrm{o}}( \sqrt{\ln n} )$ . Proposition 3.2 follows from applying (11) to $H_{n_0,n}$ .

4. Proof of Theorem 2.1

In the proof of Theorem 2.1 we will use the following lemma (see Corollary 2.3 of Fan et al. [7]), which gives self-normalized Cramér moderate deviations for martingales.

Lemma 4.1. Let $(\eta_k, \mathcal{F}_k)_{k=1,\ldots,n}$ be a finite sequence of martingale differences. Assume that there exist a constant $\rho \in (0, 1]$ and numbers $\gamma_n>0$ and $\delta_n\geq 0$ satisfying $\gamma_n, \delta_n \rightarrow 0$ such that for all $1\leq i\leq n$ ,

(12) \begin{equation} \mathbb{E}[ |\eta_{k}|^{2+\rho} \mid \mathcal{F}_{k-1 } ] \leq \gamma_n^\rho \mathbb{E}[ \eta_{k} ^2 \mid \mathcal{F}_{k-1 } ]\end{equation}

and

(13) \begin{equation} \Biggl| \sum_{k=1}^n \mathbb{E}[ \eta_{k} ^2 \mid \mathcal{F}_{k-1 } ]-1\Biggr| \leq \delta_n^2 \quad {a.s.}\end{equation}

Denote

\[V_n= \dfrac{\sum_{k=1}^n \eta_{k} }{ \sqrt{\sum_{k=1}^n \eta_{k}^2} }\]

and

\[\widehat{\gamma}_n(x, \rho) = \dfrac{ \gamma_n^{ \rho(2-\rho)/4 } }{ 1+ x ^{ \rho(2+\rho)/4 }}.\]

1. (i) If $\rho \in (0, 1)$ , then for all $0\leq x = {\textrm{o}}( \gamma_n^{-1} )$ ,

   \begin{equation*}\biggl|\ln \dfrac{\mathbb{P}(V_n \geq x)}{1-\Phi(x)} \biggr| \leq C_{\rho} \bigl( x^{2+\rho} \gamma_n^\rho+ x^2 \delta_n^2 +(1+x)( \delta_n + \widehat{\gamma}_n(x, \rho) ) \bigr) .\end{equation*}

2. (ii) If $\rho =1$ , then for all $0\leq x = {\textrm{o}}( \gamma_n^{-1} )$ ,

   \begin{equation*}\biggl|\ln \dfrac{\mathbb{P}(V_n \geq x)}{1-\Phi(x)} \biggr| \leq C \bigl( x^{3} \gamma_n + x^2 \delta_n^2+(1+x)( \delta_n+ \gamma_n |\!\ln \gamma_n| +\widehat{\gamma}_n(x, 1) ) \bigr) .\end{equation*}

Now we are in a position to prove Theorem 2.1. Denote

\[\hat{\xi}_{k+1}= \sqrt{Z_{ k}} ( Z_{ k+1}/Z_{ k} -m ),\]

$\mathcal{F}_{n_0} =\{ \emptyset, \Omega \} $ and $\mathcal{F}_{k+1}=\sigma \{ Z_{i}\colon n_0\leq i\leq k+1 \}$ for all $k\geq n_0$ . Notice that $X_{k,i}$ is independent of $Z_k$ . Then it is easy to verify that

\begin{align*} \mathbb{E}[ \hat{\xi}_{k+1} \mid \mathcal{F}_{k } ] &= Z_{ k} ^{-1/2} \mathbb{E}[ Z_{ k+1} -mZ_{ k} \mid \mathcal{F}_{k } ]\notag \\ &= Z_{ k} ^{-1/2} \sum_{i=1}^{Z_k} \mathbb{E}[ X_{ k, i} -m \mid \mathcal{F}_{k } ] \notag \\& = Z_{ k} ^{-1/2} \sum_{i=1}^{Z_k} \mathbb{E}[ X_{ k, i} -m ] \notag \\& =0.\end{align*}

Thus $(\hat{\xi}_k, \mathcal{F}_k)_{k=n_0+1,\ldots,n_0+n}$ is a finite sequence of martingale differences. Notice that $X_{ k, i}-m$ , $i\geq 1$ , are centered and independent random variables. Thus the following equalities hold:

\begin{align*} \sum_{k=n_0}^{n_0+n-1} \mathbb{E}[ \hat{\xi}_{k+1}^2 \mid \mathcal{F}_{k } ] &=\sum_{k=n_0}^{n_0+n-1} Z_{k}^{-1} \mathbb{E}[ ( Z_{ k+1} -mZ_{ k} )^2 \mid \mathcal{F}_{k } ]\notag \\& = \sum_{k=n_0}^{n_0+n-1} Z_{k}^{-1} \mathbb{E}\Biggl[ \Biggl( \sum_{i=1}^{Z_k} (X_{ k, i} -m) \Biggr)^2 \biggm|\mathcal{F}_{k } \Biggr] \notag \\&=\sum_{k=n_0}^{n_0+n-1} Z_{k}^{-1} Z_k \mathbb{E}[ (X_{ k, i} -m)^2 ] \notag \\&= n v^2. \end{align*}

Moreover, it is easy to see that

(14) \begin{align} \mathbb{E}[ |\hat{\xi}_{k+1}|^{2+\rho} \mid \mathcal{F}_{k } ] &= Z_{k}^{-1-\rho/2} \mathbb{E}[ | Z_{ k+1} -mZ_{ k} |^{2+\rho } \mid \mathcal{F}_{k } ] \notag \\[3pt] & = Z_{k}^{-1-\rho/2} \mathbb{E}\Biggl[ \Biggl| \sum_{i=1}^{Z_k} (X_{ k, i} -m) \Biggr|^{2+\rho } \biggm| \mathcal{F}_{k } \Biggr] . \end{align}

By Rosenthal’s inequality, we have

\begin{align*} \mathbb{E}\Biggl[ \Biggl| \sum_{i=1}^{Z_k} (X_{ k, i} -m) \Biggr|^{2+\rho } \biggm|\mathcal{F}_{k } \Biggr]&\leq C^{\prime}_\rho \Biggl(\Biggl( \sum_{i=1}^{Z_k}\mathbb{E}( X_{ k, i} -m )^2 \Biggr)^{1+\rho/2} + \sum_{i=1}^{Z_k}\mathbb{E}| X_{ k, i} -m |^{2+\rho } \Biggr) \\[3pt] & \leq C^{\prime}_\rho \bigl( Z_k^{1+\rho/2} v^{2+\rho} + Z_k \mathbb{E}| Z_1 -m |^{2+\rho } \bigr).\end{align*}

Since the set of extinction of the process $(Z_{k})_{k\geq 0}$ is negligible with respect to the annealed law $\mathbb{P}$ , we have $Z_k\geq 1$ for any k. From (14), by the last inequality and the fact $Z_k\geq 1$ , we deduce that

\begin{align} \mathbb{E}[ |\hat{\xi}_{k+1}|^{2+\rho} \mid \mathcal{F}_{k } ] &\leq C^{\prime}_\rho \bigl( v^{ \rho} + \mathbb{E}| Z_1 -m |^{2+\rho }/v^2 \bigr) v^2 \notag \\[3pt] & = C^{\prime}_\rho \bigl( v^{ \rho} + \mathbb{E}| Z_1 -m |^{2+\rho }/v^2 \bigr) \mathbb{E}[ \hat{\xi}_{k+1}^{2} \mid \mathcal{F}_{k } ]\notag \\[3pt] &\leq C^{\prime}_\rho \bigl( v^{ \rho} + 2^{1+\rho} \bigl(\mathbb{E} Z_1 ^{2+\rho } + m ^{2+\rho }\bigr)/v^2 \bigr) \mathbb{E}[ \hat{\xi}_{k+1}^{2} \mid \mathcal{F}_{k } ] . \notag\end{align}

By Jensen’s inequality, we have $ m ^{2+\rho } = (\mathbb{E} Z_1) ^{2+\rho } \leq \mathbb{E} Z_1 ^{2+\rho }. $ Thus we have

\begin{equation*} \mathbb{E}[ |\hat{\xi}_{k+1}|^{2+\rho} \mid \mathcal{F}_{k } ] \leq C_\rho \bigl( v^{ \rho} + \mathbb{E} Z_1 ^{2+\rho }/v^2 \bigr) \mathbb{E}[ \hat{\xi}_{k+1}^{2} \mid \mathcal{F}_{k } ]. \end{equation*}

Let $\eta_k =\hat{\xi}_{n_0+k}/(\sqrt{n} v)$ and $\mathcal{F}_{k}=\mathcal{F}_{n_0+k}$ . Then $(\eta_k, \mathcal{F}_{k})_{k=1,\ldots,n}$ is a martingale difference sequence and satisfies conditions (12) and (13) with $ \delta_n=0$ and

\[ \gamma_n=\bigl( C_\rho \bigl( v^{ \rho} + \mathbb{E} Z_1 ^{2+\rho }/v^2 \bigr) \bigr)^{1/\rho}/ {(\sqrt{n} v)}. \]

Clearly,

\[M_{n_0,n}= \dfrac{ \sum_{k=1}^{n} \eta_{k } }{\sqrt{\sum_{k=1}^{n} \eta_{k }^2 } }.\]

Applying Lemma 4.1 to $(\eta_k, \mathcal{F}_{k})_{k=1,\ldots,n}$ , we obtain the desired inequalities. Notice that for any $\rho \in (0, 1]$ and all $x\geq 0$ , the following inequality holds:

\[ \dfrac{ 1+x}{ 1+ x ^{ \rho(2+\rho)/4 }} \leq C_\rho (1+x)^{1-\rho(2+\rho)/4}. \]

5. Proof of Corollary 2.1

We only give a proof of Corollary 2.1 for $\rho \in (0, 1)$ . The proof for $\rho=1$ is similar. We first show that for any Borel set $B\subset \mathbb{R}$ ,

(15) \begin{equation}\limsup_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl(\dfrac{M_{n_0,n} }{ a_n } \in B \biggr) \leq - \inf_{x \in \overline{B}}\dfrac{x^2}{2}.\end{equation}

When $B =\emptyset$ , the last inequality is obvious, with $-\inf_{x \in \emptyset}{{x^2}/{2}}=-\infty$ . Thus we may assume that $B \neq \emptyset$ . Let $x_0=\inf_{x\in B} |x|$ . Clearly, we have $x_0\geq\inf_{x\in \overline{B}} |x|$ . Then, by Theorem 2.1, it follows that for $\rho \in (0, 1)$ and $a_n ={\textrm{o}}(\sqrt{n})$ ,

\begin{align*} \mathbb{P}\biggl(\dfrac{ M_{n_0,n} }{ a_n } \in B \biggr) &\leq \mathbb{P}( | M_{n_0,n}| \geq a_n x_0)\\ & \leq 2( 1-\Phi ( a_nx_0)) \exp\biggl\{ C_\rho \biggl( \dfrac{ ( a_nx_0)^{2+\rho} }{n^{\rho/2}} + \dfrac{ (1+a_nx_0)^{1-\rho(2+\rho)/4} }{n^{\rho(2-\rho)/8}} \biggr) \biggr\}.\end{align*}

Using the inequalities

(16) \begin{equation}\dfrac{1}{\sqrt{2 \pi}(1+x)} {\textrm{e}}^{-x^2/2} \leq 1-\Phi ( x ) \leq \dfrac{1}{\sqrt{ \pi}(1+x)} {\textrm{e}}^{-x^2/2}, \quad x\geq 0,\end{equation}

and the fact that $a_n \rightarrow \infty$ and $a_n/\sqrt{n}\rightarrow 0$ , we obtain

\begin{equation*}\limsup_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl(\dfrac{ M_{n_0,n} }{ a_n } \in B \biggr) \leq -\dfrac{x_0^2}{2} \leq - \inf_{x \in \overline{B}}\dfrac{x^2}{2} ,\end{equation*}

which gives (15).

Next we prove that

(17) \begin{equation}\liminf_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl(\dfrac{ M_{n_0,n} }{ a_n } \in B \biggr) \geq - \inf_{x \in B^o}\dfrac{x^2}{2} .\end{equation}

When $B^o =\emptyset$ , the last inequality is obvious, with $ -\inf_{x \in \emptyset}{{x^2}/{2}}=-\infty$ . Thus we may assume that $B^o \neq \emptyset$ . Since $B^o$ is an open set, for any given small $\varepsilon_1>0$ there exists an $x_0 \in B^o$ such that

\begin{equation*} 0< \dfrac{x_0^2}{2} \leq \inf_{x \in B^o}\dfrac{x^2}{2} +\varepsilon_1.\end{equation*}

Again by the fact that $B^o$ is an open set, for $x_0 \in B^o$ and all sufficiently small $\varepsilon_2 \in (0, |x_0|] $ , we have $(x_0-\varepsilon_2, x_0+\varepsilon_2] \subset B^o$ . Without loss of generality, we may assume that $x_0>0$ . Clearly, we have

(18) \begin{align}\mathbb{P}\biggl(\dfrac{M_{n_0,n} }{ a_n } \in B \biggr) &\geq \mathbb{P}( M_{n_0,n} \in (a_n ( x_0-\varepsilon_2), a_n( x_0+\varepsilon_2)] ) \notag \\&= \mathbb{P}( M_{n_0,n} \geq a_n ( x_0-\varepsilon_2) )-\mathbb{P}( M_{n_0,n} \geq a_n( x_0+\varepsilon_2) x). \end{align}

Again by Theorem 2.1, it is easy to see that for $a_n \rightarrow \infty$ and $ a_n ={\textrm{o}}(\sqrt{n} )$ ,

\[\lim_{n\rightarrow \infty} \dfrac{\mathbb{P}(M_{n_0,n} \geq a_n( x_0+\varepsilon_2) ) }{\mathbb{P}( M_{n_0,n} \geq a_n ( x_0-\varepsilon_2) ) } =0 .\]

From (18), by the last line and Theorem 2.1, for all n large enough and $a_n ={\textrm{o}}(\sqrt{n} )$ it holds that

\begin{align*}\mathbb{P}\biggl(\dfrac{M_{n_0,n} }{ a_n } \in B \biggr) & \geq \dfrac12 \mathbb{P}( M_{n_0,n} \geq a_n ( x_0-\varepsilon_2) ) \\& \geq \dfrac12 ( 1-\Phi ( a_n( x_0-\varepsilon_2))) \exp\biggl\{ -C_\rho \biggl( \dfrac{ ( a_nx_0)^{2+\rho} }{n^{\rho/2}} + \dfrac{ (1+a_nx_0)^{1-\rho(2+\rho)/4} }{n^{\rho(2-\rho)/8}} \biggr) \biggr\}.\end{align*}

Using (16) and the fact that $a_n \rightarrow \infty$ and $a_n/\sqrt{n}\rightarrow 0$ , after some calculations we get

\begin{equation*} \liminf_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl(\dfrac{M_{n_0,n} }{ a_n } \in B \biggr) \geq - \dfrac{1}{2}( x_0-\varepsilon_2)^2 . \end{equation*}

Letting $\varepsilon_2\rightarrow 0$ , we deduce that

\begin{equation*}\liminf_{n\rightarrow \infty}\dfrac{1}{a_n^2}\ln \mathbb{P}\biggl(\dfrac{ M_{n_0,n} }{ a_n } \in B \biggr) \geq - \dfrac{x_0^2}{2} \geq -\inf_{x \in B^o}\dfrac{x^2}{2} -\varepsilon_1.\end{equation*}

Since $\varepsilon_1$ can be arbitrarily small, we get (17). Combining (15) and (17), we complete the proof of Corollary 2.1.

6. Proof of Theorem 2.2

Recall the martingale differences $(\eta_k, \mathcal{F}_{k})_{k=1,\ldots,n }$ defined in the proof of Theorem 2.1. Then $\eta_k$ satisfies conditions (12) and (13) with $\delta_n=0$ and

\[\gamma_n=\bigl( C_\rho \bigl( v^{ \rho} + \mathbb{E} Z_1 ^{2+\rho }/v^2 \bigr) \bigr)^{1/\rho}/ \sqrt{n} v.\]

Clearly, we have $H_{n_0,n}= \sum_{k=1}^{n} \eta_k$ . Applying Theorem 2.1 of Fan [5] to $(\eta_k, \mathcal{F}_{k})_{k=1,\ldots,n}$ , we obtain the desired inequalities.