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Limit theorems for delayed sums of random sequence

Journal of Inequalities and Applications20122012:124

https://doi.org/10.1186/1029-242X-2012-124

• Accepted: 31 May 2012
• Published:

Abstract

For a sequence of arbitrarily dependent random variables (X n )nNand Borel sets (B n )nN, on real line the strong limit theorems, represented by inequalities, i.e. the strong deviation theorems of the delayed average ${S}_{n.{k}_{n}}\left(\omega \right)$ are investigated by using the notion of asymptotic delayed log-likelihood ratio. The results obtained popularizes the methods proposed by Liu.

Mathematics Subject Classification 2000: Primary, 60F15.

Keywords

• strong deviation theorem
• likelihood ratio
• delayed sums

1. Introduction

Let (a n )nNbe a sequence of real numbers and let (k n )nNbe a sequence of positive integers. The numbers
${\rho }_{n,{k}_{n}}=\left\{\sum _{j=1}^{{k}_{n}}{a}_{n+j-1}\right\}/{k}_{n}$

are called the (forward) delayed first arithmetic means (See ). In , using the limiting behavior of delayed average, Chow found necessary and sufficient conditions for the Borel summability of i.i.d. random variables and also obtained very simple proofs of a number of well-known results such as the Hsu-Robbins-Spitzer-Katz theorem. In , Lai studied the analogues of the law of the iterated logarithim for delayed sums of independent random variables. Recently, Chen  has presented an accurate description the limiting behavior of delayed sums under a non-identically distribution setup, and has deduced Chover-type laws of the iterated logarithm for them.

Our aim in this article is to establish strong deviation theorems (limit theorem expressed by inequalities, see ) of delayed average for the dependent absolutely continuous random variables. By using the notion of asymptotic delayed log-likelihood ratio, we extend the analytic technique proposed by Liu  to the case of delayed sums. The crucial part of the proof is to construct a delayed likelihood ratio depending on a parameter, and then applies the Borel-Cantelli lemma.

Throughout, let (X n )nNbe a sequence of absolutely continuous random variables on a fixed probability space $\left\{\Omega ,\mathsc{F},P\right\}$ with the joint density function g1, n(x1,..., x n ), n N, and f j (x), j = 1, 2,... be the the marginal density function of random variable X j . (k n )nNbe a subsequence of positive integers, such that, for every ε > 0, ${\sum }_{n=1}^{\infty }\mathsf{\text{exp}}\left(-{k}_{n}\epsilon \right)<\infty$.

Definition 1. The delayed likelihood ratio is defined by
${\mathsc{L}}_{n}\left(\omega \right)=\left\{\begin{array}{cc}\frac{{\Pi }_{j=n}^{n+{k}_{n}-1}{f}_{j}\left({X}_{j}\right)}{{{g}^{n,n+{k}_{n}-1}}^{{}_{\left({X}_{n},...,{X}_{n+{}_{kn}-1}\right)}}},\hfill & \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{denominator}}\phantom{\rule{2.77695pt}{0ex}}>0\hfill \\ 0,\hfill & \mathsf{\text{otherwise}}\hfill \end{array}\right\$
(1.1)
Let
$\mathsc{L}\left(\omega \right)=-\underset{n}{\mathsf{\text{lim}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{inf}}}\phantom{\rule{0.3em}{0ex}}\frac{1}{{k}_{n}}\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)$
(1.2)

$\mathsc{L}\left(\omega \right)$ is called asymptotic delayed log-likelihood ratio, where ${g}^{n,n+{k}_{n}-1}\left({x}_{n},\dots ,{{x}_{n}}_{+{k}_{n}-1}\right)$ denotes the joint density function of random vector $\left({X}_{n},\dots ,{X}_{n+{k}_{n}-1}\right)$, ω is a sample point (with log 0 = -∞).

It will be shown in Lemma 1 that $\mathsc{L}\left(\omega \right)\ge 0$a.e. in any case.

Remark 1. It will be seen below that $\mathsc{L}\left(\omega \right)$ has the analogous properties of the likelihood ratio in , Although $\mathsc{L}\left(\omega \right)$ is not a proper metric among probability measures, we nevertheless consider it as a measure of "discrimination" between the dependence (their joint distribution) and independence (the product of their marginals). Obviously, ${\mathsc{L}}_{n}\left(\omega \right)=1$, a.e. n N if (X n )nNis independent. In view of the above discussion of the asymptotic logarithmic delayed likelihood ratio, it is natural for us to think of $\mathsc{L}\left(\omega \right)$ as a measure how far (the random deviation) of (X n )nNis from being independent and how dependent they are. The closer $\mathsc{L}\left(\omega \right)$ approaches to 0, the smaller the deviation is.

Lemma 1. Let${\mathsc{L}}_{n}\left(\omega \right)$be define as above, then
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)\le 0,\phantom{\rule{2.77695pt}{0ex}}a.e.$
(1.3)
Proof. Let $B=\left\{\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right):{g}^{n,n+{k}_{n}-1}\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right)>0\right\}.$ Since
$\begin{array}{c}E\left[{\mathsc{L}}_{n}\left(\omega \right)\right]\hfill \\ =\int \cdots \underset{\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right)\in B}{\int }\frac{{\prod }_{j=n}^{n+{k}_{n}-1}{f}_{j}\left({x}_{j}\right)}{{g}^{n,n+{k}_{n}-1}\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right)}\hfill \\ \phantom{\rule{1em}{0ex}}.{g}^{n,n+{k}_{n}-1}\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right)d{x}_{n}\dots d{x}_{n+{k}_{n}-1}\hfill \\ =\int \cdots \underset{\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right)\in B}{\int }\prod _{j=n}^{n+{k}_{n}-1}{f}_{j}\left({x}_{j}\right)d{x}_{n}\dots d{x}_{n+{k}_{n}-1}\hfill \\ \le \int \cdots \underset{\left({x}_{n},\dots ,{x}_{n+{k}_{n}-1}\right)\in {R}^{{k}_{n}}}{\int }\prod _{j=n}^{n+{k}_{n}-1}{f}_{j}\left({x}_{j}\right)d{x}_{n}\dots d{x}_{n+{k}_{n}-1}=1.\hfill \end{array}$
From Markov inequality, for every ε > 0, we have
$P\phantom{\rule{0.5em}{0ex}}\left[\frac{1}{{k}_{n}}\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)\ge \epsilon \right]=P\phantom{\rule{0.2em}{0ex}}\left[{\mathsc{L}}_{n}\left(\omega \right)\ge \mathsf{\text{exp}}\left({k}_{n}\epsilon \right)\right]\le 1\cdot \mathsf{\text{exp}}\left(-{k}_{n}\epsilon \right).$
Hence
$\sum _{n=1}^{\infty }P\phantom{\rule{0.5em}{0ex}}\left[\frac{1}{{k}_{n}}\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)\ge \epsilon \right]\le \sum _{n=1}^{\infty }\mathsf{\text{exp}}\left(-{k}_{n}\epsilon \right)<\infty .$
By Borel-Cantelli lemma, we have
$P\phantom{\rule{0.5em}{0ex}}\left[\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)\ge 2\epsilon \right]=0,$

for any ε > 0, (1.3) follows immediately. □

2. Main results and proofs

Theorem 1. Let (X n )nN, ${\mathsc{L}}_{n}\left(\omega \right)$, $\mathsc{L}\left(\omega \right)$be defined as above, (B n )nNbe a sequence of Borel sets of the real line. Let ${S}_{n,{k}_{n}}\left(\omega \right)=\frac{1}{{k}_{n}}{\sum }_{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)$, and assume
$c=\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\sum _{j=n}^{n+{k}_{n}-1}P\left({X}_{j}\in {B}_{j}\right),$
(2.1)
then
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\left(\omega \right)\le {\left(\sqrt{\mathsc{L}\left(\omega \right)}+\sqrt{c}\right)}^{2},\phantom{\rule{2.77695pt}{0ex}}a.e.$
(2.2)

where${1}_{{B}_{n}}\left(\cdot \right)$be the indicator function of B n .

Proof. Assume s > 0 to be a constant, and let
${h}_{j}\left({x}_{j}\right)=\frac{{{s}^{{1}_{{B}_{j}}}}^{\left({x}_{j}\right)}{f}_{j}\left({x}_{j}\right)}{1+\left(s-1\right){\int }_{{B}_{j}}{f}_{j}\left({x}_{j}\right)d{x}_{j}},\phantom{\rule{1em}{0ex}}j=1,2,\dots$
(2.3)
It is not difficult to see that $\int {h}_{j}\left({x}_{j}\right)d{x}_{j}=1$, j = 1, 2,... Let
${\Lambda }_{n}\left(s,\omega \right)=\left\{\begin{array}{cc}\frac{{\Pi }_{j=n}^{n+{k}_{n}-1}{h}_{j}\left({X}_{j}\right)}{{g}^{n,n+{k}_{n}-1}\left({{X}_{n}}_{,...,}{X}_{n+{k}_{n}-1}\right)},\hfill & \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{denominator}}\phantom{\rule{2.77695pt}{0ex}}>0\hfill \\ 0,\hfill & \mathsf{\text{otherwise}}\hfill \end{array}\right\$
(2.4)
From Lemma 1, there exists $A\left(s\right)\in \mathsc{F}$, P(A(s)) = 1, such that
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\mathsf{\text{log}}{\Lambda }_{n}\left(s,\omega \right)\le 0,\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.5)
Since ${\int }_{{B}_{j}}{f}_{j}\left({x}_{j}\right)d{x}_{j}=P\left({X}_{j}\in {B}_{j}\right)$, by (2.3) we have
$\begin{array}{c}\prod _{j=n}^{n+{k}_{n}-1}{h}_{j}\left({x}_{j}\right)\hfill \\ =\prod _{j=n}^{n+{k}_{n}-1}\frac{{s}^{{1}_{{B}_{j}}\left({x}_{j}\right)}{f}_{j}\left({x}_{j}\right)}{1+\left(s-1\right){\int }_{{B}_{j}}{f}_{j}\left({x}_{j}\right)d{x}_{j}}\hfill \\ ={s}^{{\sum }_{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({x}_{j}\right)}\prod _{j=n}^{n+{k}_{n}-1}\frac{{f}_{j}\left({x}_{j}\right)}{1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)}\hfill \end{array}$
(2.6)
It follows from (1.1), (2.4) and (2.6) that
$\mathsf{\text{log}}{\Lambda }_{n}\left(s,\omega \right)=\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)\mathsf{\text{log}}s-\sum _{j=n}^{n+{k}_{n}-1}\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]+\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)$
(2.7)
(2.5) and (2.7) yield
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\mathsf{\text{log}}s\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)-\sum _{j=n}^{n+{k}_{n}-1}\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]+\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)\right)\le 0,\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.8)
Let s > 1, dividing the two sides of (2.8) by log s, we have
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)-\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-\mathsf{\text{1}}\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}+\frac{\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)}{\mathsf{\text{log}}s}\right)\le 0,\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.9)
By (1.2), (2.9) and the property lim sup n (a n - b n ) ≤ d lim sup n (a n - c n ) ≤ lim sup n (b n - c n ) + d, one gets
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)-\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-\mathsf{\text{1}}\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}\right)\le \frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s},\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.10)
By (2.10) and the property of the superior above and the inequality 0 < log(1+x) ≤ x(x > 0), we obtain
$\begin{array}{c}\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\left(\omega \right)\hfill \\ \le \underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}\right)+\frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s}\hfill \\ \le \underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}\frac{\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)}{\mathsf{\text{log}}s}\right)+\frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s}\hfill \\ \le c\left(\frac{s-1}{\mathsf{\text{log}}s}\right)+\frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s},\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)\hfill \end{array}$
(2.11)
(2.11) and the inequality $1-\frac{1}{s}<\mathsf{\text{log}}s\left(s>1\right)$ imply
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\left(\omega \right)\le c\cdot s+\frac{s\mathsc{L}\left(\omega \right)}{s-1},\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.12)
Let D be a set of countable real numbers dense in the interval (1, +∞), and let A* = ∩sDA(s), g(s, x) = cs + sx/(s - 1), then we have by (2.12)
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\left(\omega \right)\le g\left(s,\mathsc{L}\left(\omega \right)\right),\phantom{\rule{1em}{0ex}}\omega \in {A}^{*},\phantom{\rule{1em}{0ex}}s\in D$
(2.13)
Let c > 0, it easy to see that if $\mathsc{L}\left(\omega \right)>0$, a.e., then, for fixed ω, $g\left(s,\mathsc{L}\left(\omega \right)\right)$as a function of s attains its smallest value $g\left(1+\sqrt{\mathsc{L}\left(\omega \right)/c},\mathsc{L}\left(\omega \right)\right)=2\sqrt{c\mathsc{L}\left(\omega \right)}+\mathsc{L}\left(\omega \right)+c$ on the interval (1, +∞), and g(s, 0) is increasing on the interval (1, +∞) and lims→1+ g(s, 0) = 0. For each ω A* ∩ A(1), if $\mathsc{L}\left(\omega \right)\ne \infty$, take κ n (ω) D, n = 1, 2,..., such that ${\kappa }_{n}\left(\omega \right)\to 1+\sqrt{\mathsc{L}\left(\omega \right)/c}$. We have by the continuity of $g\left(s,\mathsc{L}\left(\omega \right)\right)$ with respect to s,
$\underset{n\to +\infty }{\mathsf{\text{lim}}}g\left({\kappa }_{n}\left(\omega \right),\mathsc{L}\left(\omega \right)\right)={\left(\sqrt{\mathsc{L}\left(\omega \right)}+\sqrt{c}\right)}^{2},$
(2.14)
By (2.13), we obtain
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\le g\left({\kappa }_{n}\left(\omega \right),\phantom{\rule{2.77695pt}{0ex}}\mathsc{L}\left(\omega \right)\right),\phantom{\rule{1em}{0ex}}n=1,2,\dots$
(2.15)
(2.14) and (2.15) imply
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\left(\omega \right)\le {\left(\sqrt{\mathsc{L}\left(\omega \right)}+\sqrt{c}\right)}^{2},\phantom{\rule{1em}{0ex}}\omega \in {A}^{*}\cap A\left(1\right)$
(2.16)

If $\mathsc{L}\left(\omega \right)=\infty$, (2.16) holds trivially. Since P (A* ∩ A(1)) = 1, (2.2) holds by (2.16), when c > 0.

When c = 0, we have by letting s = e in (2.11),
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}{S}_{n,{k}_{n}}\left(\omega \right)\le \mathsc{L}\left(\omega \right),\phantom{\rule{1em}{0ex}}\omega \in A\left(e\right)$
(2.17)

since P (A(e)) = 1, (2.2) also holds by (2.17) when c = 0. □

Theorem 2. Let (X n )nN, ${\mathsc{L}}_{n}\left(\omega \right)$, $\mathsc{L}\left(\omega \right)$, (B n )nN, ${S}_{n,{k}_{n}}\left(\omega \right)$be defined as in Theorem 1 and assume
${c}^{\prime }=\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}\frac{1}{{k}_{n}}\sum _{j=n}^{n+{k}_{n}-1}P\left({X}_{j}\in {B}_{j}\right),$
(2.18)
then, if$0\le \mathsc{L}\left(\omega \right)\le {c}^{\prime }$a.e., then
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)\ge {c}^{\prime }-2\sqrt{{c}^{\prime }\mathsc{L}\left(\omega \right),\phantom{\rule{1em}{0ex}}a.e.}$
(2.19)
Proof. Let 0 <s < 1, dividing the two sides of (2.8) by log s, we have
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)-\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}+\frac{\mathsf{\text{log}}{\mathsc{L}}_{n}\left(\omega \right)}{\mathsf{\text{log}}s}\right)\ge 0,\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.20)
By (1.2), (2.20) and the property lim inf n (a n - b n ) ≥ d lim inf n (a n - c n ) ≥ lim inf n (b n - c n ) + d, one gets
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)-\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}\right)\ge \frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s},\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.21)
By (2.21) and the property of the inferior above and the inequality log(1+ x) ≤ x(-1 <x ≤ 0), we obtain
$\begin{array}{c}\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)\hfill \\ \ge \underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}\right)+\frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s}\hfill \\ \ge \underset{n}{\mathsf{\text{lim}}}\mathsf{\text{inf}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}\frac{\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)}{\mathsf{\text{log}}s}\right)+\frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s}\hfill \\ \ge {c}^{\prime }\left(\frac{s-1}{\mathsf{\text{log}}s}\right)+\frac{\mathsc{L}\left(\omega \right)}{\mathsf{\text{log}}s},\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)\hfill \end{array}$
(2.22)
(2.22) and the inequality $1-\frac{1}{s}<\mathsf{\text{log}}s$ and log s < s - 1 (0 <s < 1) imply
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)\ge {c}^{\prime }\cdot s+\frac{\mathsc{L}\left(\omega \right)}{s-1},\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)\cap A\left(1\right)$
(2.23)
Let D' be a set of countable real numbers dense in the interval (0, 1), and let ${A}_{*}={\cap }_{s\in {D}^{\prime }}A\left(s\right)$, h(s, x) = c's + x/(s - 1), then we have by (2.23)
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)\ge h\left(s,\mathsc{L}\left(\omega \right)\right),\phantom{\rule{1em}{0ex}}\omega \in {A}_{*},\phantom{\rule{1em}{0ex}}s\in {D}^{\prime }$
(2.24)
Let c' > 0, it easy to see that if $0<\mathsc{L}\left(\omega \right)<{c}^{\prime }$, a.e., then, for fixed ω, $h\left(s,\mathsc{L}\left(\omega \right)\right)$ as a function of s attains its maximum value $h\left(1-\sqrt{\mathsc{L}\left(\omega \right)/{c}^{\prime }},\mathsc{L}\left(\omega \right)\right)={c}^{\prime }-2\sqrt{{c}^{\prime }\mathsc{L}\left(\omega \right)}$, on the interval (0, 1), and h(s, 0) is increasing on the interval (0, 1) and lims→1+ h(s, 0) = c'. For each ω A*A(1), if $\mathsc{L}\left(\omega \right)\ne \infty$, take l n (ω) D', n = 1, 2,..., such that ${l}_{n}\left(\omega \right)\to 1-\sqrt{\mathsc{L}\left(\omega \right)/{c}^{\prime }}$. We have by the continuity of $h\left(s,\mathsc{L}\left(\omega \right)\right)$ with respect to s,
$\underset{n\to +\infty }{\mathsf{\text{lim}}}h\left({l}_{n}\left(\omega \right),\mathsc{L}\left(\omega \right)\right)={c}^{\prime }-2\sqrt{{c}^{\prime }\mathsc{L}\left(\omega \right)},$
(2.25)
By (2.24), we obtain
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\ge h\left({l}_{n}\left(\omega \right),\mathsc{L}\left(\omega \right)\right),\phantom{\rule{1em}{0ex}}n=1,2,\dots$
(2.26)
(2.25) and (2.26) imply
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)\ge {c}^{\prime }-2\sqrt{{c}^{\prime }\mathsc{L}\left(\omega \right)},\phantom{\rule{1em}{0ex}}\omega \in {A}_{*}\cap A\left(1\right)$
(2.27)

If $\mathsc{L}\left(\omega \right)=\infty$, (2.27) holds trivially. Since P (A*A(1)) = 1, (2.19) holds by (2.27), when c' > 0. (2.19) also holds trivially when c' = 0. □

Remark 2. In case $\mathsc{L}\left(\omega \right)>{c}^{\prime }\ge 0$, a.e., we cannot get a better lower bound of $\mathsf{\text{lim}}\underset{n}{\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)$. This motivates the following problem: under the conditions of Theorem 2, how to get a better lower bound of $\mathsf{\text{lim}}\underset{n}{\mathsf{\text{inf}}}{S}_{n,{k}_{n}}\left(\omega \right)$ in case of $\mathsc{L}\left(\omega \right)>{c}^{\prime }\ge 0$, a.e.?

Definition 2. (Generalized empirical distribution function) Let (X n )nNbe identically distribution with common distribution function F , for each m, n N, let
${F}_{m,n}\left(x\right)=\frac{1}{n}\sum _{k=m}^{m+n-1}{1}_{\left({X}_{k}\le x\right)}.$

F m, n = the observed frequency of values that are ≤ x from time m to m + n - 1. The F1,nis the usual empirical distribution function, hence the name given above.

In particular, let B = (-∞, x], x R in Theorems 1 and 2, we can get a strong limit theorem for the generalized empirical distribution function.

Corollary 1. Let (X n )nNbe i.i.d. random variables with common distribution function F, let B n = (-∞, x], n = 1, 2,..., then
$\underset{n}{\mathsf{\text{lim}}}{F}_{n,n+{k}_{n}-1}\left(x\right)=F\left(x\right),\phantom{\rule{2.77695pt}{0ex}}a.e.$
Corollary 2. Let (X n )nNbe independent random variables and (B n )nNbe as Theorem 1, then
$\underset{n}{\mathsf{\text{lim}}}\frac{1}{{k}_{n}}\sum _{j=n}^{n+{k}_{n}-1}\left[{1}_{{B}_{j}}\left({X}_{j}\right)-P\left({X}_{j}\in {B}_{j}\right)\right]=0,\phantom{\rule{2.77695pt}{0ex}}a.e.$
(2.28)
Proof. Note that P (X j B j ) ≤ 1, j = 1, 2,... and in this case, 0 ≤ c, c' ≤ 1, $\mathsc{L}\left(\omega \right)=0$a.e., we have by (2.11)
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}{1}_{{B}_{j}}\left({X}_{j}\right)-\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}\right)\le 0,\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)$
(2.29)
by (2.29) and the property of the superior above and the inequality 0 ≤ log(1+x) ≤ x(x > 0), we obtain
$\begin{array}{c}\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\sum _{j=n}^{n+{k}_{n}-1}\left[{1}_{{B}_{j}}\left({X}_{j}\right)-P\left({X}_{j}\in {B}_{j}\right)\right]\hfill \\ \le \underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}\frac{\mathsf{\text{log}}\left[1+\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)\right]}{\mathsf{\text{log}}s}-P\left({X}_{j}\in {B}_{j}\right)\right)\hfill \\ \le \underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\left(\sum _{j=n}^{n+{k}_{n}-1}\frac{\left(s-1\right)P\left({X}_{j}\in {B}_{j}\right)}{\mathsf{\text{log}}s}-P\left({X}_{j}\in {B}_{j}\right)\right)\hfill \\ \le \left(\frac{s-1}{\mathsf{\text{log}}s}-1\right),\phantom{\rule{1em}{0ex}}\omega \in A\left(s\right)\hfill \end{array}$
(2.30)
Analogously as in the proof of Theorem 1, we obtain
$\underset{n}{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{1}{{k}_{n}}\sum _{j=n}^{n+{k}_{n}-1}\left[{1}_{{B}_{j}}\left({X}_{j}\right)-P\left({X}_{j}\in {B}_{j}\right)\right]\le 0,\phantom{\rule{2.77695pt}{0ex}}a.e.$
(2.31)

Similarly, we have $\mathsf{\text{lim}}{\mathsf{\text{inf}}}_{n}\frac{1}{{k}_{n}}\sum _{j=n}^{n+{k}_{n}-1}\left[{1}_{{B}_{j}}\left({X}_{j}\right)-P\left({X}_{j}\in {B}_{j}\right)\right]\ge 0$, a.e. hence (2.28) follows immediately. □

Remark 3. Let B n = B, Corollary 2 implies that $\frac{\mathsf{\text{lim}}{S}_{n,{k}_{n}}}{{k}_{n}}=P\left({X}_{1}\in B\right)$ which gives the strong law of large numbers for the delayed arithmatic means.

Declarations

Acknowledgements

This work is supported by The National Natural Science Foundation of China (Grant No. 11071104) and the An Hui University of Technology research grant: D2011025. The authors would like to thank two referees for their insightful comments which resulted in improving Theorems 1, 2 and Corollary 2 significantly.

Authors’ Affiliations

(1)
Department of Mathematics & Physics, HeFei University, HeFei, 230601, P. R. China
(2)
Faculty of Mathematics & Physics, AnHui University of Technology, Ma'anshan, 243002, P. R. China

References 