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Strong law of large numbers under a general moment condition. (English) Zbl 1112.60024
Summary: We use our maximum inequality for \(p\)th order random variables \((p>1)\) to prove a strong law of large numbers (SLLN) for sequences of \(p\)th order random variables. In particular, in the case \(p=2\) our result shows that \(\sum f(k)/k< \infty\) is a sufficient condition for SLLN for \(f\)-quasi-stationary sequences to hold. It was known that the above condition, under the additional assumption of monotonicity of \(f\), implies SLLN [P. Erdős, Trans. Am. Math. Soc. 67, No. 1, 51–56 (1949; Zbl 0034.07201); I. Gal and J. Koksma, Nederl. Akad. Wet., Proc. 53, 638–653 and Indag. Math. 12, 192–207 (1950; Zbl 0041.02406); V. F. Gaposhkin, Theory Probab. Appl. 22, 286–310 (1977), translation from Teor. Veroyatn. Primen. 22, 295–319 (1977; Zbl 0377.60033); F. Moricz, Z. Wahrscheinlichkeitstheorie Verw. Geb. 38, No. 3, 223–236 (1977; Zbl 0336.60027)]. Besides getting rid of the monotonicity condition, the inequality enables us to extend the general result to \(p\)th order random variables, as well as to the case of Banach-space-valued random variables.

MSC:
60F15 Strong limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F25 \(L^p\)-limit theorems
60G10 Stationary stochastic processes
60G12 General second-order stochastic processes
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