On the Distribution of Sums of Independent Random Variables

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 Levy 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 Levy’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.