Generalization of the steinhaus problem on the convergence of series with random signs

Abstract

For series of random variables$\sum\limits_{k = 1}^\infty {a_k x_k }$,a _K ∈R ¹, {X _K } ^∞ _K=1 being an Ising system, i.e., for each n ≥ 2 the joint distribution of {X _K } ⁿ _K=1 has the form

$$P_n (t_1 ,...,t_n ) = ch^{ - (n - 1)} J \cdot exp(J\sum\limits_{k - 1}^{n - 1} {t_k t_{k + 1} )\prod\limits_{k = 1}^n {\frac{1}{2}\delta (t_{k^{ - 1} }^2 ),J > 0} }$$

one obtains a criterion for almost everywhere convergence:$\sum\limits_{k = 1}^\infty {a_k^2< \infty }$. The relation between the asymptotic behavior of large deviations of the sum and the rate of decrease of the sequence {a_k} of the coefficients is investigated.