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.     

A note on the sum of uniform random variables

An inductive procedure is used to obtain distributions and probability densities for the sum S_n of independent, non-equally uniform random variables. Some known results are then shown to follow immediately as special cases. Under the assumption of equally uniform random variables some new formulas are obtained for probabilities and means related to S_n. Finally, some new recursive formulas involving distributions are derived.     

On the clustering of independent uniform random variables

We consider the number K_n of clusters at a distance level d_n ∈ (0, 1) of n independent random variables uniformly distributed in [0, 1], or the number K_n of connected components in the random interval graph generated by these variables and d_n, and, depending upon how fast d_n → 0 as n → ∞, determine the asymptotic distribution of K_n, with rates of convergence, and of related random variables that describe the cluster sizes. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Vague convergence of sums of independent random variables

A sequence (μ _n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ _n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ ₀ denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e ⁻¹] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x ₁ + ... +x _n)/a _n, a_n → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ ₀. If furthermore the ratiosa _n+1/a _n are bounded above and below by positive numbers, thenL(x)=P[|X ₁|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea _n → ∞ so that the limit measure=βδ ₀. To the memory of Shlomo Horowitz This research was partially supported by the National Science Foundation.     

Symmetric sum and symmetric product of two independent random variables

The symmetry of the sum and product of two independent random variablesX andY, whenX orY is not symmetric, is studied.     

On the normal approximation of a sum of a random number of independent random variables

In this paper, we extend the results obtained in [J. Sunklodas, Some estimates of normal approximation for the distribution of a sum of a random number of independent random variables, Lith. Math. J., 52(3):326?C333, 2012] for a thrice-differentiable function h : ? ?? ? to the case of h ?? BL(?); namely, we estimate the quantity | E h(Z _N )? E h(Y)| where h is a real bounded Lipschitz function, $ {Z_N}={{{\left( {{S_N}-\mathrm{E}{S_N}} \right)}} \left/ {{\sqrt{{\mathrm{D}{S_N}}}}} \right.} $ , S _N = X ₁ + · · · + X _N , X ₁ , X ₂ , . . . are independent, not necessarily identically distributed, real random variables, N is a positive integer-valued r.v. independent of X ₁ , X ₂ , . . . , and Y is a standard normal random variable.     

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.     

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).     

On the convergence in law of almost all sums of independent random variables

The paper deals with sums of independent and identically distributed random variables defined on some probability space which are multiplied by random coefficients. These coefficients are the values of independent random variables defined on another probability space. We obtain conditions for the weak convergence of weighted sums, for almost all coefficients, to some infinitely divisible distribution. The limit distribution for these sums is found. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

A criterion of convergence of nonrandomly centered random sums of independent identically distributed random variables

Necessary and sufficient conditions are presented for the weak convergence of random sums of independent identically distributed random variables in the double array scheme. As corollaries, two criteria of the normal convergence of random sums are given. Supported by the Russian Foundation for Fundamental Research (grant No. 96-011-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.