A maximal theorem of Hardy–Littlewood type for pairwise i.i.d. random variables and the law of large numbers

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].