Book Review: Sums of independent random variables

Estimating Sums of Independent Random Variables

The paper deals with a problem proposed by Uriel Feige in 2005: if X1, . . . , Xn is a set of independent nonnegative random variables with expectations equal to 1, is it true that P ( ∑n i=1 Xi < n + 1) > 1 e ? He proved that P ( ∑n i=1Xi < n + 1) > 1 13 . In this paper we prove that infimum of the P ( ∑n i=1Xi < n + 1) can be achieved when all random variables have only two possible values, a...

Strong Laws for Weighted Sums of Negative Dependent Random Variables

In this paper, we discuss strong laws for weighted sums of pairwise negatively dependent random variables. The results on i.i.d case of Soo Hak Sung [9] are generalized and extended.

On large deviations of sums of independent random variables

Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér’s condition. The large deviation x-region under consideration is broader than in the classical Cramér’s theorem, and the estimate of the remainder is uniform with respect to x. The corresponding asymptotic expansion with...

Almost sure convergence of weighted sums of independent random variables

Let (Ω,F ,P) be a probability space, and let {Xn} be a sequence of integrable centered i.i.d. random variables. In this paper we consider what conditions should be imposed on a complex sequence {bn} with |bn| → ∞, in order to obtain a.s. convergence of P n Xn bn , whenever X1 is in a certain class of integrability. In particular, our condition allows us to generalize the rate obtained by Marcin...

Tail sums of convergent series of independent random variables