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A Maximal Theorem of Hardy--Littlewood Type for Pairwise I.I.D. Random Variables and the Law of Large Numbers
Theory of Probability and Its Applications ( IF 0.5 ) Pub Date : 2021-11-02 , DOI: 10.1137/s0040585x97t990502
T. Nguyen , H. Pham

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 445-454, January 2021.
Let $p\in [1,2)$. We show that if $(X_n)_{n=1}^\infty$ is a sequence of pairwise i.i.d. random variables with ${E}|X_1|^p<\infty$, then ${P}\{\sup_n|{S_n}/{n^{1/p}}|> \alpha\}\le {C_p\,{E}|X_1|^p}/{\alpha^p}$ for every $\alpha>0$ for some constant $C_p$ depending only on $p$, where $S_n:=\sum_{i=1}^n(X_i-{E}X_i)$. This will be proved as a consequence of a more general result where, instead of being pairwise i.i.d., the sequence $(X_n)$ is only required to be weakly correlated in the sense of E. Rio. In fact, we prove an inequality that gives the rates of the convergence $\lim_{n\to\infty}|S_n|/{n^{{1}/{p}}}=0$ a.s. and thus strengthen the main result of [E. Rio, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), pp. 469--474].


中文翻译:

成对IID随机变量的Hardy--Littlewood型极大定理和大数定律

概率论及其应用,第 66 卷,第 3 期,第 445-454 页,2021 年 1 月。
让 $p\in [1,2)$。我们证明如果 $(X_n)_{n=1}^\infty$ 是具有 ${E}|X_1|^p<\infty$ 的成对 iid 随机变量序列,则 ${P}\{\sup_n |{S_n}/{n^{1/p}}|> \alpha\}\le {C_p\,{E}|X_1|^p}/{\alpha^p}$ 每$\alpha>0 $ 用于某些仅取决于 $p$ 的常量 $C_p$,其中 $S_n:=\sum_{i=1}^n(X_i-{E}X_i)$。这将被证明是一个更一般的结果的结果,其中序列 $(X_n)$ 只需要在 E. Rio 的意义上是弱相关的,而不是成对的 iid。事实上,我们证明了一个不等式,它给出了收敛速度 $\lim_{n\to\infty}|S_n|/{n^{{1}/{p}}}=0$ as 从而加强了主[E.的结果。里约,CR Acad。科学。巴黎先生。I Math., 320 (1995), pp. 469--474]。
更新日期:2021-11-09
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