The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (Xk: k = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN
which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The
results are also extended for random fields (Xkℓ: k, ℓ = 1, 2, …).
A paper by Chow  contains (i.a.) a strong law for delayed sums, such that the length of the edge of the nth window equals nα for 0 < α < 1. In this paper we consider the kind of intermediate case when edges grow like n=L(n), where L is slowly varying at infinity, thus at a higher rate than any power less than one, but not quite at a linear rate. The typical
example one should have in mind is L(n) = log n. The main focus of the present paper is on random field versions of such strong laws.
Пусть (Xk),k=1,2,... — последов ательность случайны х переменных с нулевым средним и конечной дисперсие йσk2 Пусть α>1; если (Xk) та кова, чтоE(XkXt)=0 дляpα<k<1≦(p+1)α (p,k,l=1, 2, ...) и, более того, такова, чт о
Authors:Rolando Cavazos-Cadena and Daniel Hernández-Hernández
This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically
distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central