On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

On the accuracy of the normal approximation for sums of independent random variables

Doklady Mathematics -     

On the accuracy of the normal approximation for sums of symmetric independent random variables

Doklady Mathematics -     

Some estimates of the normal approximation for independent non-identically distributed random variables

In this paper, we generalize some results of [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003] for independent identically distributed summands to to the case of independent non-identically distributed real summands. We derive the Edgeworth expansion with the first term only. Proofs are given following [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003].     

On the equivalence of limit distributions of a sum and of a maximum sum of independent random variables

We prove the equivalence of the limit distributions of an appropriately centered and normalized sum and the maximum sum of independent random variables which have finite expectations. The result is an extension of a result of Kruglov (1999).     

The distribution of the sum of independent gamma random variables

Summary The distribution of the sum ofn independent gamma variates with different parameters is expressed as a single gamma-series whose coefficients are computed by simple recursive relations.     

The uniform convergence of the sum of independent random variables

LetX _i (i=1, 2, ...) be the independent random variables on the probability space (, ,P). In this paper we will show that the necessary and sufficient condition of the uniform convergence of X _i is the convergence of m _i ⁽¹⁾ and m _i ⁽⁰⁾ , wherem _i ⁽¹⁾ , m _i ⁽⁰⁾ denote the 1-quantile and 0-quantile ofX _i respectively.     

On normal approximation of discounted and strongly mixing random variables

We estimate the difference for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z _v = B _v ⁻¹ ∑ _i=0 ^∞ v ⁱ X _i and with discount factor ν such that 0 < ν < 1. Here {X _n , n ≥ 0} is a sequence of strongly mixing random variables with , and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)^1/2). Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07.     

Limit behavior of sums of a random number of independent random variables

Lithuanian Mathematical Journal -     

Inequalities for the distribution of a sum of functions of independent random variables

Let where ₁,..., n are independent random variables and the are functions (e.g., taking the values 0 and 1). For cases when almost all the summands forming are equal to 0 with a probability close to 1, estimates from above and below are obtained for the quantity P{=0}, as well as upper estimates for the distance in variation between the distribution , and the distribution of the approximating sum of independent random variables.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 745–758, November, 1977.The author is grateful to V. G. Mikhailov for numerous discussions of the results of this paper and for his help in carrying out the tedious auxiliary calculations.