General maximal inequalities related to the strong law of large numbers

Abstract

For any sequence (ξ _n ) of random variables, we obtain maximal inequalities from which we can derive conditions for the a.s. convergence to zero of the normalized differences

$$\frac{1}{{2^n }}\left( {\mathop {\max }\limits_{2^n \leqslant k < 2^{n + 1} } \left| {\sum\limits_{i = 2^n }^k {\xi _i } } \right| - \left| {\sum\limits_{i = 2^n }^{2^{n + 1} - 1} {\xi _i } } \right|} \right).$$

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; this condition is an improvement on the well-known Móricz conditions.