Geodesic
General Maximal Inequalities Related to the Strong Law of Large Numbers
Matematičeskie zametki, Tome 81 (2007) no. 1, pp. 98-111

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For a sequence $(\xi_n)$ of random variables, we obtain maximal inequalities from which we can derive conditions for the a.s. convergence to zero of normalized differences $$ \frac{1}{2^n} \biggl(\max_{2^n\le k2^{n+1}} \biggl|\sum^k_{i=2^n}\xi_i\biggr|-\biggl|\sum_{i=2^n}^{2^{n+1}-1}\xi_i\biggr|\biggr). $$ The convergence to zero of this sequence leads to the strong law of large numbers (SLLN). In the special case of quasistationary sequences, we obtain a sufficient condition for the SLLN, which is an improvement on the well-known Móricz conditions.
Keywords: strong law of large numbers, maximal inequality, quasistationary random sequence, Banach space, Bochner measurability, Jensen's inequality. 
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Sh. Levental; H. Salehi; S. A. Chobanyan. General Maximal Inequalities Related to the Strong Law of Large Numbers. Matematičeskie zametki, Tome 81 (2007) no. 1, pp. 98-111. http://geodesic.mathdoc.fr/item/MZM_2007_81_1_a7/