Convergence rates in the law of large numbers for arrays of martingale differences

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• 作者：Shunli Hao^a ; ^{hsl2005100@163.com" class="auth_mail} ; Quansheng Liu^b ; ^{Quansheng.Liu@univ-ubs.fr" class="auth_mail}
• 关键词：Law of large numbers ; Martingale arrays ; Convergence rate ; Baum&ndash ; Katz theorem ; Maximal inequalities ; Weighted sums
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：15 September 2014
• 年：2014
• 卷：417
• 期：2
• 页码：733-773
• 全文大小：643 K

文摘

We study the convergence rates in the law of large numbers for arrays of martingale differences. For n猢?$n猢?/mo><mn>1</mn>$, let X_n1,X_n2,…$X_{n1},X_{n2},\dots$ be a sequence of real valued martingale differences with respect to a filtration {&empty;,惟}=F_n0⊂F_n1⊂F_n2⊂鈰?/span>, and set S_nn=X_n1+鈰?X_nn. Under a simple moment condition on pt>pt> for some 纬∈(1,2]$纬\in (1,2]$, we show necessary and sufficient conditions for the convergence of the series pt>pt>${\sum }_{n=1}^{\infty }蠒(n)P\{|S_{nn}|>蔚n^{伪}\}$, where 伪  , 蔚>0$蔚>0$ and 蠒   is a positive function; we also give a criterion for 蠒(n)P{|S_nn|>蔚n^伪}→0$蠒(n)P\{|S_{nn}|>蔚n^{伪}\}\to 0$. The most interesting case where 蠒   is a regularly varying function is considered with attention. In the special case where (X_nj)_j猢?${(X_{nj})}_{j猢?/mo><mn>1</mn>}$ is the same sequence (X_j)_j猢?${(X_j)}_{j猢?/mo><mn>1</mn>}$ of independent and identically distributed random variables, our result on the series pt>pt>${\sum }_{n=1}^{\infty }蠒(n)P\{|S_{nn}|>蔚n^{伪}\}$ corresponds to the theorems of Hsu, Robbins and Erdös (1947, 1949) if 伪=1$伪=1$ and 蠒(n)=1$蠒(n)=1$, of Spitzer (1956) if 伪=1$伪=1$ and 蠒(n)=1/n$蠒(n)=1/n$, and of Baum and Katz (1965) if 伪>1/2$伪>1/2$ and 蠒(n)=n^b−1$蠒(n)=n^{b-1}$ with b猢?$b猢?/mo><mn>0</mn>$. In the single martingale case (where X_nj=X_j$X_{nj}=X_j$ for all n   and j  ), it generalizes the results of Alsmeyer (1990). The consideration of martingale arrays (rather than a single martingale) makes the results very adapted in the study of weighted sums of identically distributed random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinitely many martingale differences, say pt>pt>$S_{n,\infty }={\sum }_{j=1}^{\infty }{X}_{nj}$ instead of S_nn$S_{nn}$. The obtained results improve and extend those of Ghosal and Chandra (1998). The one-sided cases and the supermartingale case are also considered.