A generalization of the Men’shov-Rademacher theorem

Abstract

For a sequence “X _n ”_n≥1 of random variables with finite second moment and a sequence “b _n ”_n≥1 of positive constants, new sufficient conditions for the almost sure convergence of Σ_n≥1 X _n /b _n are obtained and the strong law of large numbers, which states that lim _n→∞ Σ ⁿ_k=1 X _k /b _n = 0 almost surely, is proved. The results are shown to be optimal in a number of cases. In the theorems, assumptions have the form of conditions on ρ _n = sup _k (EX _k X _k+n)⁺,

$$ r_n = \mathop {\sup }\limits_k \frac{{\left( {EX_k X_{k + n} } \right)^ + }} {{\left( {EX_k^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {EX_{k + n}^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} $$

, EX ² _n , and b _n , where x ⁺ = x ∨ 0 and n ∈ ℕ.