Large Deviations for the Maximum of the Absolute Value of Partial Sums of Random Variable Sequences

Abstract

Let $\{{\xi }_i:i\ge 1\}$ be a sequence of independent, identically distributed (i.i.d. for short) centered random variables. Let $S_n={\xi }_1+\cdots +{\xi }_n$ denote the partial sums of $\{{\xi }_i\}.$ We show that sequence $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}|S_k|:n\ge 1\}$ satisfies the large deviation principle (LDP, for short) with a good rate function under the assumption that $P({\xi }_1\ge x)$ and $P({\xi }_1\le -x)$ have the same exponential decrease.

1. Introduction

Throughout this paper, on a probability space $\{\Omega ,\mathcal{F},P\}$, let $\{{\xi }_i:i\ge 1\}$ be a sequence of independent, identically distributed (i.i.d.) centered random variables that take real values. Denote the partial sums $S_n:={\sum }_{i=1}^n{\xi }_i$ of sequence $\{{\xi }_i:i\ge 1\}.$

The seminal paper of Cramér [1] motivates our work. Cramér obtained that $\{{\displaystyle \frac{1}{n}}{S}_n:n\ge 1\}$ satisfies large deviation principle (LDP) with rate function ${\Lambda }^{*}(x)$ (see Theorem 1) under finite moments, which is the famous Cramér condition, i.e., if there exists a $\delta >0$ such that $E{e}^{\lambda |X_1|}<\infty$ for all $\left|\lambda \right|<\delta$. Cramer theorem has the following form for any measurable set $B\subset \mathbb{R}$:

$$\begin{array}{cc}\hfill -\underset{x\in B^{\circ }}{inf}{\Lambda }^{*}(x)& \le \underset{n\to \infty }{lim\; inf}\frac{1}{n}logP({\displaystyle \frac{S_n}{n}}\in B)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \le \underset{n\to \infty }{lim\; sup}\frac{1}{n}logP({\displaystyle \frac{S_n}{n}}\in B)\le -\underset{x\in \overline{B}}{inf}{\Lambda }^{*}(x).\hfill \end{array}$$

(2)

where $B^{\circ }$ denotes the interior of B and $\overline{B}$ denotes its closure. We call inequality (1) the large deviations lower bound and inequality (2) the large deviations upper bound. If both hold, then sequence $\{{\displaystyle \frac{1}{n}}{S}_n:n\ge 1\}$ satisfies LDP with rate function ${\Lambda }^{*}(x)$. In other words, the theory of LDP deals with large fluctuations and the probability of such large fluctuations decays exponentially.

The tail probability $P(S_n\ge nx)$ of independent random variables was researched in detail in many papers. Nagaev [2] obtained that partial sums $\{{\displaystyle \frac{1}{n}}{S}_n:n\ge 1\}$ for i.i.d. random variables and found that it satisfies LDP under the assumption that $P({\xi }_1\ge x)$ decreases similarly to a power function. Soon, Nagaev [3] obtained the bounds for probabilities of partial sums of independent random variables, by weakening the requirement, on the hypothesis that generalized and ordinary moments are finite. Under Cramer condition, Kiesel and Stadtmuller [4] extended Cramer theorem to weighted sums of i.i.d. random variables. Moreover, Gantert, Ramanan and Rembart [5] researched the LDP for weighted sums of i.i.d. random variables with stretched exponential tails.

The tail probability $P(\underset{1\le k\le n}{max}{S}_k\ge nx)$ has been researched in depth. Under the Cramer condition, Borovkov and Korshunov [6] conducted work for time-homogeneous Markov chain and Shklyaev [7] conducted work for i.i.d. random variables; both obtained LDP. Soon after, Kozlov [8] obtained LDP results by applying a direct probability approach to $P(\underset{1\le k\le n}{max}{S}_k\ge nx)$ of i.i.d. non-degenerate random variables, which obey the Cramer condition. Lately, Fan, Grama and Liu [9] established the LDP for sequence $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}{S}_k:n\ge 1\}$ of martingale differences random variables under finite subexponential moments condition.

Feller [10] mentioned the importance of the estimation of tail probability $P(\underset{1\le k\le n}{max}|S_k|\ge nx)$, which has attracted broad attention in recent decades. Recently, Li [11] established the upper bound estimation for probability $P(\underset{1\le k\le n}{max}|S_k|\ge nx)$ of martingale differences random variables bounded in $L^p$. For strictly stationary and negatively associated random variables, Xing and Yang [12] obtained some exponential inequalities for the maximum of the absolute value of partial sums via classical techniques based on blocking and truncation. Moreover, the upper bound estimation for tail probability $P(\underset{1\le k\le n}{max}|S_k|\ge nx)$ for martingale differences random variables was obtained by Fan, Grama and Liu [13] in situations where conditional subexponential moments are bounded.

The above results demonstrate that the research for probability $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}|S_k|:n\ge 1\}$ only obtained large deviations upper bound. To fill this gap, we shall primarily obtain the result that sequence $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}|S_k|:n\ge 1\}$ of i.i.d. random variables satisfies LDP under the assumption that $P({\xi }_1\ge x)$ and $P({\xi }_1\le -x)$ have the same exponential decrease (see Corollary 1), i.e., we obtain large deviations lower bound and large deviations upper bound.

This article is organized as follows. We firstly introduce the necessary knowledge about definitions and theorems that we need in Section 2. Then, the main theorems and corollaries are presented in Section 3. Moreover, in Section 4, we provide the lemmas needed to prove the conclusions and proofs of our main results.

2. Preliminaries

Before we present our results and proofs, we introduce some definitions and theorems that can be found in cf. [14,15].

Definition 1.

(1) A function I:$\mathcal{F}\to [0,\infty ]$is called a rate function if it is non-negative and lower semicontinuous, i.e., the level sets$\{x:I(x)\le \alpha \}$are closed, for$\alpha \in \mathbb{R}$. (2) A rate function I is said to be good if, in addition, its level sets are compact.

Definition 2.

We say that a sequence of random variables$\{{\xi }_i:i\ge 1\}$satisfies LDP in$\mathbb{R}$with rate function I if I is a rate function and, for any measurable set$B\in \mathcal{B}(\mathbb{R}),$the following is the case:

$$\begin{array}{cc}-\underset{x\in B^{\circ }}{inf}I(x)\hfill & \le \underset{n\to \infty }{lim\; inf}\frac{1}{n}logP({\xi }_n\in B)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}\frac{1}{n}logP({\xi }_n\in B)\le -\underset{x\in \overline{B}}{inf}I(x),\hfill \end{array}$$

where$B^{\circ }$denotes the interior of B, and$\overline{B}$denotes its closure.

Theorem 1.

(Cramer’s theorem)Let $\{{\xi }_i:i\ge 1\}$ be a sequence of i.i.d. real value random variables on $(\Omega ,\mathcal{F},P)$. Let partial sums $S_n=\sum _{i=1}^n{\xi }_i$, and let $\Lambda (\theta )$ be log moment generating function of ${\xi }_1$, i.e., $\Lambda (\theta )=logE{e}^{\theta {\xi }_1}$, and let ${\Lambda }^{*}(x)$ be convex conjugate of Λ, i.e., ${\Lambda }^{*}(x)=\underset{\theta \in R}{sup}\{\theta \xi -\Lambda (\theta )\}$. Then, $\{{\displaystyle \frac{S_n}{n}}:n\ge 1\}$ satisfies LDP with rate function ${\Lambda }^{*}(x)$ in $\mathbb{R}$ if Λ is finite in a neighborhood of zero, i.e., for any measurable set $B\subset \mathbb{R}$ of the following:

$$\begin{array}{cc}\hfill -\underset{x\in B^{\circ }}{inf}{\Lambda }^{*}(x)& \le \underset{n\to \infty }{lim\; inf}\frac{1}{n}logP({\displaystyle \frac{S_n}{n}}\in B)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}\frac{1}{n}logP({\displaystyle \frac{S_n}{n}}\in B)\le -\underset{x\in \overline{B}}{inf}{\Lambda }^{*}(x).\hfill \end{array}$$

Theorem 2.

(Principle of the largest term)Let $a_n$ and $b_n$ be sequences in ${\mathbb{R}}^{+}$. Then, the following is the case:

$$\underset{n\to \infty }{lim\; sup}\frac{1}{n}log(a_n+b_n)\le \underset{n\to \infty }{lim\; sup}\frac{1}{n}log(a_n)\vee \underset{n\to \infty }{lim\; sup}\frac{1}{n}log(b_n),$$

and the following is the case.

$$\underset{n\to \infty }{lim\; inf}\frac{1}{n}log(a_n+b_n)\ge \underset{n\to \infty }{lim\; inf}\frac{1}{n}log(a_n)\vee \underset{n\to \infty }{lim\; inf}\frac{1}{n}log(b_n).$$

3. Main Results

Let $\{{\xi }_i:i\ge 1\}$ be a sequence of i.i.d. centered random variables and denote $S_n:={\displaystyle \sum _{i=1}^n}{\xi }_i$. Then, we shall investigate the LDP for the sequence of $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}|S_k|:n\ge 1\}$. The main results of this paper are as follows.

Theorem 3.

Let $\{{\xi }_i:i\ge 1\}$ be a sequence of i.i.d. random variables. If $E{\xi }_1=0$, $E{{\xi }_1}^2<\infty$ and for some constants $\alpha \in (0,1)$, $0<C_1\le C_2$, the following is the case:

$$-C_2\le \underset{x\to \infty }{lim\; inf}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\le \underset{x\to \infty }{lim\; sup}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\le -C_1,$$

then for all x > 0, we have the following.

$$\begin{array}{cc}\hfill -C_2{x}^{\alpha }& \le \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\le -C_1{x}^{\alpha }.\hfill \end{array}$$

Theorem 4.

Let $\{{\xi }_i:i\ge 1\}$ be a sequence of i.i.d. random variables. If $E{\xi }_1=0$ and for some constants $\alpha \in (0,1)$, $0<C_1\le C_2$, $0<C_3\le C_4$, the following is the case:

$$-C_2\le \underset{x\to \infty }{lim\; inf}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\le \underset{x\to \infty }{lim\; sup}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\le -C_1,$$

$$-C_4\le \underset{x\to \infty }{lim\; inf}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\le -x)\le \underset{x\to \infty }{lim\; sup}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\le -x)\le -C_3,$$

then for all x > 0, we have the following.

$$\begin{array}{cc}\hfill -(C_2\wedge C_4)x^{\alpha }& \le \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}|S_k|\ge nx)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}|S_k|\ge nx)\le -(C_1\wedge C_3)x^{\alpha }.\hfill \end{array}$$

Corollary 1.

Let $\{{\xi }_i:i\ge 1\}$ be a sequence of i.i.d. random variables. If $E{\xi }_i=0$, and for some constants $\alpha \in (0,1)$, $C>0$, we have the following:

$$\underset{x\to \infty }{lim}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)=-C,$$

$$\underset{x\to \infty }{lim}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\le -x)=-C,$$

then for all x > 0, the following is obtained.

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}|S_k|\ge nx)=-C{x}^{\alpha }.$$

Then, $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}|S_k|:n\ge 1\}$ satisfies LDP with the good rate function $I(x)=C{x}^{\alpha }.$

4. Proofs of Main Results

To prove our main results, we need the following lemmas, and we also will provide their proofs.

Lemma 1.

For a random variable ${\xi }_1$ with $E{\xi }_1=0$, we assume $E({{\xi }_1}^{2}exp\{{({\xi }_1^{+})}^{\alpha }\})<\infty$, for some constant $\alpha \in (0,1)$. Set ${\eta }_1={\xi }_1{1}_{\{{\xi }_1\le y\}}$, for y > 0. Then, the following is the case.

$$E{e}^{y^{\alpha -1}{\eta }_1}\le 1+\frac{y^{2\alpha -2}}{2}E({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\}).$$

Proof of Lemma 1.

By Taylor’s expansion, we can obtain

$$e^{y^{\alpha -1}{\eta }_1}\le 1+y^{\alpha -1}{\eta }_1+\frac{y^{2\alpha -2}{\eta }_1^2}{2}{e}^{y^{\alpha -1}{\eta }_1^{+}}.$$

The following is the case:

$$\begin{array}{ccc}\hfill {\eta }_1^{+}& =& {\xi }_1{1}_{\{0\le {\xi }_1\le y\}}\hfill \\ & \le & y^{1-\alpha }{{\xi }_1}^{\alpha }{1}_{\{0\le {\xi }_1\le y\}}\hfill \\ & \le & y^{1-\alpha }{({\xi }_1^{+})}^{\alpha },\hfill \end{array}$$

and ${\eta }_1^2\le {\xi }_1^2$. Then, we obtain the following.

$$\begin{array}{ccc}\hfill E{e}^{y^{\alpha -1}{\eta }_1}& \le & 1+y^{\alpha -1}E{\eta }_1+\frac{y^{2\alpha -2}}{2}E({\eta }_1^2{e}^{y^{\alpha -1}{\eta }_1^{+}})\hfill \\ & \le & 1+y^{\alpha -1}E{\xi }_1+\frac{y^{2\alpha -2}}{2}E({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\})\hfill \\ & =& 1+\frac{y^{2\alpha -2}}{2}E({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\}).\hfill \end{array}$$

Thus, we complete the proof of Lemma 1.  □

Lemma 2.

Assume $\{{\xi }_i:i\ge 1\}$ is an i.i.d. random variables sequence. If $E{\xi }_1=0$, and for some constants $\alpha \in (0,1)$, $C>0$, $E{\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\}<\infty$,

$$\underset{x\to \infty }{lim\; sup}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\le -C,$$

then for all x > 0, the following is the case.

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\le -C{x}^{\alpha }.\end{array}$$

Proof of Lemma 2.

Set ${\eta }_i={\xi }_i{1}_{\{{\xi }_i\le y\}}$ for $y>0$. Then, the following is the case.

$$\begin{array}{cc}\hfill P(\underset{1\le k\le n}{max}{S}_k\ge x)& \le P(\underset{1\le k\le n}{max}\sum _{i=1}^k{\eta }_i\ge x)+P(\underset{1\le k\le n}{max}\sum _{i=1}^k{\xi }_i{1}_{\{{\xi }_i>y\}}>0)\hfill \\ \hfill & =P(\sum _{i=1}^k{\eta }_i\ge x,\exists k\in [1,n])+P(\underset{1\le i\le n}{max}{\xi }_i>y)\hfill \\ \hfill & :=P_1+P_2.\hfill \end{array}$$

(3)

For all $x>0$, denote stopping time

$$T(x)=min\{k\in [1,n]:\sum _{i=1}^k{\eta }_i\ge x\}\mathrm{and}minØ=0.$$

We easily obtain

$$1_{\{{\sum }_{i=1}^k{\eta }_i\ge x,\exists k\in [1,n]\}}=\sum _{k=1}^n{1}_{\{T(x)=k\}}.$$

In order to obtain the upper bound of $P_1$, we consider martingale $Z(\lambda )=\{(Z_k(\lambda ),{\mathcal{F}}_k):k\ge 0\}$, where ${\mathcal{F}}_k=\sigma ({\xi }_1,{\xi }_2,\cdots ,{\xi }_k),k\ge 0,$ and the following is the case.

$$Z_k(\lambda )=\prod _{i=1}^k{\displaystyle \frac{exp\{\lambda {\eta }_i\}}{Eexp\{\lambda {\eta }_i\}}},Z_0(\lambda )=1.$$

Let the following be the case.

$$Z_{T(x)\wedge k}(\lambda )=\prod _{i=1}^{T(x)\wedge k}{\displaystyle \frac{exp\{\lambda {\eta }_i\}}{Eexp\{\lambda {\eta }_i\}}},Z_0(\lambda )=1.$$

By the property of martingale, then $\{(Z_{T(x)\wedge k}(\lambda ),{\mathcal{F}}_k):k\ge 0\}$ is also a martingale. Because $E(Z_{T(x)\wedge n}(\lambda ))=E(Z_0(\lambda ))=1,$ then we define the probability measure $d{P}_{\lambda }:=Z_{T(x)\wedge n}dP$ and define the expectation with respect to $P_{\lambda }$ by $E_{\lambda }$.

$$\begin{array}{cc}\hfill P_1& =E_{\lambda }\left[Z_{T(x)\wedge n}{(\lambda )}^{-1}{1}_{\{{\sum }_{i=1}^k{\eta }_i\ge x,\exists k\in [1,n]\}}\right]\hfill \\ \hfill & =E_{\lambda }\left[{(\prod _{i=1}^{T(x)\wedge n}{\displaystyle \frac{exp\{\lambda {\eta }_i\}}{Eexp\{\lambda {\eta }_i\}}})}^{-1}\sum _{k=1}^n{1}_{\{T(x)=k\}}\right]\hfill \\ \hfill & =\sum _{k=1}^n{E}_{\lambda }\left[{(\prod _{i=1}^{T(x)\wedge n}{\displaystyle \frac{exp\{\lambda {\eta }_i\}}{Eexp\{\lambda {\eta }_i\}}})}^{-1}{1}_{\{T(x)=k\}}\right]\hfill \\ \hfill & =\sum _{k=1}^n{E}_{\lambda }\left[{(\prod _{i=1}^k{\displaystyle \frac{exp\{\lambda {\eta }_i\}}{Eexp\{\lambda {\eta }_i\}}})}^{-1}{1}_{\{T(x)=k\}}\right]\hfill \\ \hfill & =\sum _{k=1}^n{E}_{\lambda }\left[exp\{-\sum _{i=1}^{k}log{\displaystyle \frac{exp\{\lambda {\eta }_i\}}{Eexp\{\lambda {\eta }_i\}}}\}{1}_{\{T(x)=k\}}\right]\hfill \\ \hfill & =\sum _{k=1}^n{E}_{\lambda }\left[exp\{-\lambda \sum _{i=1}^k{\eta }_i+\sum _{i=1}^{k}logE{e}^{\lambda {\eta }_i}\}{1}_{\{T(x)=k\}}\right]\hfill \\ \hfill & =\sum _{k=1}^n{E}_{\lambda }\left[exp\{-\lambda \sum _{i=1}^k{\eta }_i+klogE{e}^{\lambda {\eta }_1}\}{1}_{\{T(x)=k\}}\right].\hfill \end{array}$$

(4)

Under the conditions of Lemma 2, we take $\lambda =y^{\alpha -1}$, and by Lemma 1, and $log(1+t)\le t$, for $\forall t\ge 0$, we obtain the following.

$$\begin{array}{cc}\hfill logE{e}^{y^{\alpha -1}{\eta }_1}& \le log(1+\frac{y^{2\alpha -2}}{2}E({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\}))\hfill \\ \hfill & \le \frac{y^{2\alpha -2}}{2}E({{\xi }_1}^{2}exp\{{({{\xi }_1}^{+})}^{\alpha }\}).\hfill \end{array}$$

(5)

On the set $\{T(x)=k\}$, we obtain ${\sum }_{i=1}^k{\eta }_i\ge x$. Combining this fact with (4) and (5), we obtain that, for all $x>0$, the following is the case.

$$\begin{array}{cc}\hfill P_1& \le \sum _{k=1}^n{E}_{\lambda }(exp\{-\lambda x+n\frac{y^{2\alpha -2}}{2}E({{\xi }_1}^{2}exp\{{({{\xi }_1}^{+})}^{\alpha }\})\}{1}_{\{T(x)=k\}})\hfill \\ \hfill & \le exp\{-y^{\alpha -1}x+n\frac{y^{2\alpha -2}}{2}E({{\xi }_1}^{2}exp\{{({{\xi }_1}^{+})}^{\alpha }\})\}{E}_{\lambda }(\sum _{k=1}^n{1}_{\{T(x)=k\}})\hfill \\ \hfill & \le exp\{-y^{\alpha -1}x+n\frac{y^{2\alpha -2}}{2}E({{\xi }_1}^{2}exp\{{({{\xi }_1}^{+})}^{\alpha }\})\}.\hfill \end{array}$$

(6)

Next, using the Markov inequality, we obtain the following.

$$\begin{array}{cc}\hfill P_2& =P(\bigcup _{i=1}^n\{{\xi }_i>y\})\hfill \\ \hfill & \le nP({\xi }_1>y)\hfill \\ \hfill & \le nP({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\}>y^{2}exp\{y^{\alpha }\})\hfill \\ \hfill & \le \frac{n}{y^2}exp\{-y^{\alpha }\}E({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\}).\hfill \end{array}$$

(7)

Let $y=x.$ Combining (3), (6) and (7) together, we obtain the following.

$$\begin{array}{cc}\hfill & P(\underset{1\le k\le n}{max}{S}_k\ge x)\hfill \\ \hfill \le & exp\{-x^{\alpha }+{\displaystyle \frac{nE({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\})}{2x^{2-2\alpha }}}\}+{\displaystyle \frac{nE({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\})}{x^2}}{e}^{-x^{\alpha }}\hfill \\ \hfill =& e^{-x^{\alpha }}(exp\{{\displaystyle \frac{nE({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\})}{2x^{2-2\alpha }}}\}+{\displaystyle \frac{nE({\xi }_1^{2}exp\{{({\xi }_1^{+})}^{\alpha }\})}{x^2}}).\hfill \end{array}$$

Now, we replace x by nx in the above inequality; then, the following is obtained.

$$\begin{array}{cc}\hfill P(\underset{1\le k\le n}{max}{S}_k\ge nx)& \le e^{-n^{\alpha }{x}^{\alpha }}(exp\{{\displaystyle \frac{E({{\xi }_1}^{2}exp\{{({{\xi }_1}^{+})}^{\alpha }\})}{2n^{1-2\alpha }{x}^{2-2\alpha }}}\}+{\displaystyle \frac{E({{\xi }_1}^{2}exp\{{({{\xi }_1}^{+})}^{\alpha }\})}{n{x}^2}}).\hfill \end{array}$$

Take log and limsup to both sides and use the principle of the largest term; here, we obtain the LDP upper bound.

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}\frac{1}{n^{\alpha }}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)& \le -x^{\alpha }.\hfill \end{array}$$

Now we end the proof of Lemma 2.  □

In the following, we prove Theorem 3.

Proof of Theorem 3.

(i) Firstly, we prove the upper bound. Let $\epsilon \in (0,1)$ be fixed, ${\xi }_1^{\prime }={C_1}^{\frac{1}{\alpha }}{\xi }_1{({\displaystyle \frac{C_1-\epsilon }{C_1}})}^{\beta }$, $\beta >\frac{1}{\alpha }.$ By the condition given in Theorem 3, we have the following:

$$\underset{x\to \infty }{lim\; sup}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\le -C_1,$$

we obtain for $\forall \epsilon >0$, $\exists x_0$, such that when $x>x_0$,

$${\displaystyle \frac{logP({\xi }_1\ge x)}{x^{\alpha }}}\le -C_1+\epsilon ;$$

that is,

$$P({\xi }_1\ge x)\le exp\{-(C_1-\epsilon )x^{\alpha }\}.$$

Thus, for $\forall \epsilon >0$, $\exists x_0$, such that when $x>x_0$,

$$P({\xi }_1^{\prime }\ge x)\le exp\{-{(C_1-\epsilon )}^{1-\alpha \beta }{C_1}^{\alpha \beta -1}{x}^{\alpha }\}.$$

Then, the following is the case:

$$\begin{array}{cc}\hfill E\{{({{\xi }_1^{\prime }}^{+})}^{2}exp\{{({{\xi }_1^{\prime }}^{+})}^{\alpha }\}\}& ={\int }_0^{\infty }P({\xi }_1^{{}^{\prime }}\ge x)(2x+\alpha x^{\alpha -1})e^{x^{\alpha }}\hfill \\ \hfill & \le 2{\int }_0^{\infty }x{e}^{-\theta x^{\alpha }}dx+\alpha {\int }_0^{\infty }{x}^{\alpha +1}{e}^{-\theta x^{\alpha }}dx\hfill \\ \hfill & <\infty ,\hfill \end{array}$$

where $\theta ={(C_1-\epsilon )}^{1-\alpha \beta }{C_1}^{\alpha \beta -1}-1$, $\theta >0.$

Because $E{\xi }_1^2<\infty$, one can easily obtain $E{({\xi }_1^{{}^{\prime }})}^2=C_1^{\frac{2}{\alpha }}{({\displaystyle \frac{C_1-\epsilon }{C_1}})}^{2\beta }E{\xi }_1^2<\infty .$ Then, we obtain the following.

$$\begin{array}{cc}\hfill E{({\xi }_1^{\prime })}^{2}exp\{{({{\xi }_1^{\prime }}^{+})}^{\alpha }\}& =E({({\xi }_1^{\prime })}^{2}exp\{{({{\xi }_1^{\prime }}^{+})}^{\alpha }\}{1}_{\{{\xi }_1^{\prime }\ge 0\}}+{({\xi }_1^{\prime })}^{2}exp\{{({{\xi }_1^{\prime }}^{+})}^{\alpha }\}{1}_{\{{\xi }_1^{\prime }<0\}})\hfill \\ \hfill & =E({({\xi }_1^{\prime })}^{2}exp\{{({{\xi }_1^{\prime }}^{+})}^{\alpha }\}{1}_{\{{\xi }_1^{\prime }\ge 0\}}+{({\xi }_1^{\prime })}^2{1}_{\{{\xi }_1^{{}^{\prime }}<0\}})\hfill \\ \hfill & \le E{({{\xi }_1^{\prime }}^{+})}^{\alpha }exp\{{({{\xi }_1^{\prime }}^{+})}^{\alpha }\}+E{({\xi }_1^{\prime })}^2\hfill \\ \hfill & <\infty .\hfill \end{array}$$

Thus, sequence $\{{\xi }_i^{\prime }\}$ satisfies the conditions of Lemma 2, and we denote $S_k^{\prime }={\displaystyle \sum _{i=1}^k}{\xi }_i^{\prime }$. Then, we obtain, for all $x>0,$ the following.

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; sup}\frac{1}{n^{\alpha }}logP(\underset{1\le k\le n}{max}{S}_k^{\prime }\ge nx)\hfill \\ \hfill =& \underset{n\to \infty }{lim\; sup}\frac{1}{n^{\alpha }}logP({C_1}^{\frac{1}{\alpha }}{(1-\frac{\epsilon }{C_1})}^{\beta }\underset{1\le k\le n}{max}{S}_k\ge nx)\hfill \\ \hfill \le & -x^{\alpha }.\hfill \end{array}$$

Thus, we obtain the following.

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}\frac{1}{n^{\alpha }}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)& \le -C_1{(1-\frac{\epsilon }{C_1})}^{\alpha \beta }{x}^{\alpha }.\hfill \end{array}$$

Letting $\epsilon \to 0$, we obtain the following.

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}\frac{1}{n^{\alpha }}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)& \le -C_1{x}^{\alpha }.\hfill \end{array}$$

(8)

(ii) Next, we will prove the lower bound.

Because $\{{\xi }_i:i\ge 1\}$ is an i.i.d sequence, the following is the case.

$$\begin{array}{cc}\hfill P(\underset{1\le k\le n}{max}{S}_k\ge nx)& \ge P(S_n\ge nx)\hfill \\ \hfill & =P({\xi }_1+\sum _{i=2}^n{\xi }_i\ge n(\epsilon +x)-n\epsilon )\hfill \\ \hfill & \ge P(\{\sum _{i=2}^n{\xi }_i\ge -n\epsilon \}\cap \{{\xi }_1\ge n(\epsilon +x)\})\hfill \\ \hfill & =P(\sum _{i=2}^n{\xi }_i\ge -n\epsilon )P({\xi }_1\ge n(\epsilon +x)).\hfill \end{array}$$

(9)

By using the weak law of large numbers and the following fact:

$$\{\sum _{i=2}^n{\xi }_i\ge -(n-1)\epsilon \}\subseteq \{\sum _{i=2}^n{\xi }_i\ge -n\epsilon \},$$

we know the following.

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}P(\sum _{i=2}^n{\xi }_i\ge -n\epsilon )=1.\end{array}$$

(10)

Then, by the condition in Theorem 3, $\underset{x\to \infty }{lim\; inf}{\displaystyle \frac{1}{x^{\alpha }}}logP({\xi }_1\ge x)\ge -C_2,$ we obtain $\forall \epsilon >0,\exists x_0,s.t.\forall x>x_0$,

$${\displaystyle \frac{logP({\xi }_1\ge x)}{x^{\alpha }}}\ge -C_2-\epsilon .$$

Then, the following is the case.

$$P({\xi }_1\ge x)\ge exp\{-(C_2+\epsilon )x^{\alpha }\}.$$

Thus, we obtain the following.

$$\begin{array}{c}\hfill P({\xi }_1\ge n(x+\epsilon ))\ge exp\{-{\left[(x+\epsilon )n\right]}^{\alpha }(C_2+\epsilon )\}.\end{array}$$

(11)

Combining (9), (10) and (11) together, we easily obtain the following.

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; inf}\frac{1}{n^{\alpha }}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\hfill \\ \hfill \ge & \underset{n\to \infty }{lim\; inf}\frac{1}{n^{\alpha }}logP(\sum _{i=2}^n{\xi }_i\ge -n\epsilon )+\underset{n\to \infty }{lim\; inf}\frac{1}{n^{\alpha }}logP({\xi }_1\ge n(\epsilon +x))\hfill \\ \hfill \ge & -(C_2+\epsilon ){(x+\epsilon )}^{\alpha }.\hfill \end{array}$$

Letting $\epsilon \to 0$, we obtain the following.

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}\frac{1}{n^{\alpha }}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)& \ge -C_2{x}^{\alpha }.\hfill \end{array}$$

(12)

At last, by (8) and (12), we obtain, for all x > 0, the following.

$$\begin{array}{cc}\hfill -C_2{x}^{\alpha }& \le \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\le -C_1{x}^{\alpha }.\hfill \end{array}$$

Thus, we complete the proof of Theorem 3.  □

In the following, we prove Theorem 4.

Proof of Theorem 4.

By the condition $\underset{x\to \infty }{lim\; inf}\frac{1}{x^{\alpha }}logP({\xi }_1\ge x)\le -C_1,$ we obtain for $\forall \epsilon >0,\exists x_0,$ such that when $\forall x>x_0,$ the following.

$$P({\xi }_1\ge x)\le exp\{-(C_1-\epsilon )x^{\alpha }\}.$$

By condition $\underset{x\to \infty }{lim\; inf}\frac{1}{x^{\alpha }}logP({\xi }_1\le -x)\le -C_3,$ we obtain for $\forall \epsilon >0,\exists x_0,$ such that when $\forall x>x_0,P({\xi }_1\le -x)\le exp\{-(C_3-\epsilon )x^{\alpha }\}.$

Thus, we obtain the following.

$$\begin{array}{cc}\hfill E{{\xi }_1}^2& ={\int }_0^{\infty }2xP(|{\xi }_1|\ge x)dx\hfill \\ \hfill & ={\int }_0^{x_0}2xP(|{\xi }_1|\ge x)dx+{\int }_{x_0}^{\infty }2xP(|{\xi }_1|\ge x)dx\hfill \\ \hfill & \le {x_0}^2+{\int }_{x_0}^{\infty }2xP({\xi }_1\ge x)dx+{\int }_{x_0}^{\infty }2xP({\xi }_1\le -x)dx\hfill \\ \hfill & \le {x_0}^2+{\int }_{x_0}^{\infty }2xexp\{-(C_1-\epsilon )x^{\alpha }\}dx+{\int }_{x_0}^{\infty }2xexp\{-(C_3-\epsilon )x^{\alpha }\}dx\hfill \\ \hfill & <\infty .\hfill \end{array}$$

Thus, by Theorem 3, we obtain the following:

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\le -C_1{x}^{\alpha },$$

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\le -nx)\le -C_3{x}^{\alpha },$$

and we know the following.

$$P(\underset{1\le k\le n}{max}|S_k|\ge nx)\le P(\underset{1\le k\le n}{max}{S}_k\ge nx)+P(\underset{1\le k\le n}{max}(-S_k)\ge nx).$$

Thus we obtain the following inequality by the principle of the largest term.

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}|S_k|\ge nx)\hfill \\ \hfill \le & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}log(P(\underset{1\le k\le n}{max}{S}_k\ge nx)+P(\underset{1\le k\le n}{max}(-S_k)\ge nx))\hfill \\ \hfill \le & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\vee \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}(-S_k)\ge nx))\hfill \\ \hfill \le & (-C_1{x}^{\alpha })\vee (-C_3{x}^{\alpha })\hfill \\ \hfill \le & -(C_1\wedge C_3)x^{\alpha }.\hfill \end{array}$$

(13)

By the given conditions in Theorem 4, $\underset{x\to \infty }{lim\; inf}\frac{1}{x^{\alpha }}logP({\xi }_1\ge x)\ge -C_2,\underset{x\to \infty }{lim\; inf}\frac{1}{x^{\alpha }}logP({\xi }_1\le -x)\ge -C_4,$ we obtain the following.

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\ge -C_2{x}^{\alpha },$$

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\le -nx)\ge -C_4{x}^{\alpha }.$$

Since the following is the case:

$$P(\underset{1\le k\le n}{max}|S_k|\ge nx)\le P(\underset{1\le k\le n}{max}{S}_k\ge nx)\vee P(\underset{1\le k\le n}{max}(-S_k)\ge nx),$$

then we obtain the following.

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}|S_k|\ge nx)\hfill \\ \hfill & \ge \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}log(P(\underset{1\le k\le n}{max}{S}_k\ge nx)\vee P(\underset{1\le k\le n}{max}(-S_k)\ge nx))\hfill \\ \hfill & \ge \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}{S}_k\ge nx)\vee \underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}(-S_k)\ge nx))\hfill \\ \hfill & \ge (-C_2{x}^{\alpha })\vee (-C_4{x}^{\alpha })\hfill \\ \hfill & \ge -(C_2\wedge C_4)x^{\alpha }.\hfill \end{array}$$

(14)

Combining (13) and (14), we complete the proof of Theorem 4.  □

Proof of Corollary 1.

Take $C_1=C_2=C_3=C_4=C$ in Theorem 4, we can obtain the following easily for all x > 0.

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n^{\alpha }}}logP(\underset{1\le k\le n}{max}|S_k|\ge nx)=-C{x}^{\alpha }.$$

Because the upper bound and the lower bound are same, we can obtain the fact that $\{{\displaystyle \frac{1}{n}}\underset{1\le k\le n}{max}|S_k|:n\ge 1\}$ satisfies LDP with good rate function $I(x)=C{x}^{\alpha }.$  □

5. Conclusions

We obtained LDP for the maximum of the absolute value of partial sums of i.i.d. centered random variables under the assumption that $P({\xi }_1\ge x)$ and $P({\xi }_1\le -x)$ have the same exponential decrease. For further research, we will consider LDP for the maximum of the absolute value of partial sums of other types of dependent random variables, such as martingale differences and acceptable random variables.