Convergence of Numerical Characteristics of Sums of Independent Random Variables with Values in a Hilbert Space

Convergence of moments and related functional in the general central limit theorem in banach spaces

“…Theorem 3.3 generalizes to B-valued triangular arrays a result of Kruglov [10] for the Hilbert space case. Our approach is different; it may be mentioned that Kruglov's methods do not seem to extend readily to the Banach space case.…”

Convergence of weighted sums of independent random variables

“…In order that for suitably chosen constants A. the distribution functions of the sums (1.4) converge to the stable laws G = G(7, cl, c2, a), 0 < a < 2, it is necessary and sufficient that the following conditionsbe satisfied: 1/a el We need a condition for convergence of moments of sums (1.4). The following theorem is a particular case of the general result of Kruglov [11]. Let Xk and X~ be independent and identically distributed random variables.…”

Strong law of large numbers for weighted sums

“…We begin with the definition of a special class of functions, which is a part of the one introduced in [2]. Denote by g the class of continuous real functions ~o defined on the real line R satisfying the following properties:…”

“…(i) ~(z) [oz asz Ic~; (2) ~o is even, 9~(z) = ~(-x), :~ E R; (3) there exists a constant u > 1 such that ~p(z + y) <_ ap(x)~'(9) for any x, !/E R.…”

“…In [2], it was proved that if ~, E g, then ~o(z) < a exp{/3]zt} for some er > 0 and/3 > 0. It is easy to verify that the following functions belong to the class g: ~(x) = exp{/3Ixp} for N > 0 and 0 < a <_ 1;…”