General maximal inequalities related to the strong law of large numbers |
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Authors: | S Levental H Salehi S A Chobanyan |
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Institution: | 1. Michigan State University, USA 2. Muskhelishvili Institute of Computational Mathematics, Academy of Sciences, Georgia, Russia
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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. |
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