A maximal theorem of Hardy–Littlewood type for pairwise i.i.d. random variables and the law of large numbers
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 552-564
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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 $\mathbf{E}|X_1|^p<\infty$, then $\mathbf{P}\{\sup_n|{S_n}/{n^{1/p}}|> \alpha\}\le {C_p\,\mathbf{E}|X_1|^p}/{\alpha^p}$ for every $\alpha>0$ for some constant $C_p$ depending only on $p$, where $S_n:=\sum_{i=1}^n(X_i-\mathbf{E}X_i)$. 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].
Keywords:
pairwise i.i.d., the law of large numbers, Hardy–Littlewood maximal theorem.
@article{TVP_2021_66_3_a7,
author = {T. Nguyen and H. Pham},
title = {A~maximal theorem of {Hardy{\textendash}Littlewood} type for pairwise i.i.d. random variables and the law of large numbers},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {552--564},
year = {2021},
volume = {66},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a7/}
}
TY - JOUR AU - T. Nguyen AU - H. Pham TI - A maximal theorem of Hardy–Littlewood type for pairwise i.i.d. random variables and the law of large numbers JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2021 SP - 552 EP - 564 VL - 66 IS - 3 UR - http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a7/ LA - ru ID - TVP_2021_66_3_a7 ER -
%0 Journal Article %A T. Nguyen %A H. Pham %T A maximal theorem of Hardy–Littlewood type for pairwise i.i.d. random variables and the law of large numbers %J Teoriâ veroâtnostej i ee primeneniâ %D 2021 %P 552-564 %V 66 %N 3 %U http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a7/ %G ru %F TVP_2021_66_3_a7
T. Nguyen; H. Pham. A maximal theorem of Hardy–Littlewood type for pairwise i.i.d. random variables and the law of large numbers. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 552-564. http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a7/
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