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 ${E}|X_1|^p \alpha\}\le {C_p\,{E}|X_1|^p}/...