On the Distribution of Sums of Independent Random Variables

Abstract

Let {X _j ; j = 1, 2,...} be a finite sequence of independent random variables. Let S = Σ X _j be their sum, and let P _j be the distribution of X _j . Let M be the measure defined on the line deprived of its origin by M (A) = Σ _j P _j {A ∩ {0}^c}. The purpose of the present paper is to develop certain results on the approximation of the distribution L (S)of S by the accompanying infinitely divisible distribution which has for Paul Lévy measure the measure M itself. If λ= || M || is the total mass of M then V = M/λ is a probability measure. Let {Z _k ; k = 1, 2,...} be an independent sequence of random variables having common distribution V. Let N be a Poisson variable independent of the Z _k and such that EN = λ. A “natural” infinitely divisible approximation to the distribution of S is the distribution of T = Zk with Z _o = 0. If μ is a signed measure, let || μ || be its norm, equal to the total mass || μ || = ||μ ⁺ || + || u ⁻ ||. It can be shown that in some cases the approximation of L (S)by L (T)is good in the sense that || L (S) — L (T) || is small. More generally it will be shown that the Kolmogorov-Smirnov distance ϱ [L (S), L (T)] is small. This distance is defined by

$$ \left( \mu ,\upsilon \right)=\sup \left| \mu \left\{ \left( -\infty ,x \right] \right\}-\upsilon \left\{ \left( -\infty ,x \right] \right\} \right| $$

for any two signed measures,u and v. One could also use Paul Lévy’s diagonal distance Λ [μ, v] defined as the infimum of numbers α such that

$$ v\left\{ \left( -\infty ,x-\alpha \right] \right\}-\alpha \le \mu \left\{ \left( -\infty ,x \right] \right\}\le v\left\{ \left[ -\infty ,x+\alpha \right] \right\}+\alpha $$

for every value of x. However, since Λ is not invariant under scale changes, approximations in this sense are not always entirely satisfactory.

This paper was prepared with the partial support of the United States Army Research Office (Durham), grant DA-ARO(D)-31-124-G 83.