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Article Dans Une Revue Journal of Mathematical Analysis and Applications Année : 2014

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

Résumé

We study the convergence rates in the law of large numbers for arrays of martingale differences. For $n\geq 1$, let $X_{n1}$, $X_{n2}$ $\cdots$, be a sequence of real valued martingale differences with respect to a filtration $\{\emptyset, \Omega\}=\cal F_{n0}\subset\cal F_{n1}\subset\cal F_{n2} \subset\cdots $, and set $S_{nn}=X_{n1}+\cdots+X_{nn}$. Under a simple moment condition on $\sum_{j=1}^n E[|X_{nj}|^\gamma|\cal F_{n,j-1}]$ for some $\gamma\in(1,2]$, we show necessary and sufficient conditions for the convergence of the series $\sum_{n=1}^{\infty}\phi(n)P\{|S_{nn}|>\varepsilon n^\alpha\}$,where $\alpha$, $\varepsilon>0$ and $\phi$ is a positive function; we also give a criterion for $\phi(n)P\{|S_{nn}|>\varepsilon n^\alpha\} \rightarrow 0$.The most interesting case where $\phi$ is a regularly varying function is considered with attention.%$\phi(n)=n^{b-1}l(n)$ with $b\geq 0$ and $l(\cdot)>0$ slowly varying at $\infty$.In the special case where $(X_{nj})_{j\geq 1}$ are the same sequence $(X_{j})_{j\geq 1}$ of independent and identically distributed random variables, our result on the series $\sum_{n=1}^{\infty}\phi(n)P\{|S_{nn}|>\varepsilon n^\alpha\}$ corresponds to the theorems of Hsu-Robbins-Erd\"{o}s (1947, 1949) if $\alpha=1$ and $\phi(n)=1$, of Spitzer(1956) if $\alpha=1$ and $\phi(n)=1/n $, and of Baum and Katz (1965) if $\alpha>1/2$ and $\phi(n)=n^{b-1}$ with $b\geq 0$. In the single martingale case (where $X_{nj}=X_j$ 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 $S_{n, \infty}=\sum_{j=1}^\infty X_{nj}$ instead of $S_{nn}$. The obtained results improve and extend those of Ghosal and Chandra (1998). The one-sided cases and the supermartingale case are also considered.

Dates et versions

hal-01095079 , version 1 (15-12-2014)

Identifiants

Citer

Shunli Hao, Quansheng Liu. Convergence rates in the law of large numbers for arrays of martingale differences.. Journal of Mathematical Analysis and Applications, 2014, 417 (2), pp.733-773. ⟨10.1016/j.jmaa.2014.03.049⟩. ⟨hal-01095079⟩
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