Identifiability of distributions of independent random variables by linear combinations and moments

Abstract

Let X₁, X₂, ..., X_n be independent random variables. Given the moments EX_j^s (s=1, 2,...,m), (j=1, 2,...,n), the joint distribution function of the linear forms Y_i=Σ_j=1ⁿa_ijX_j, i=1, 2,...,k with an arbitrary nonvanishing joint characteristic function uniquely determines the distributions of X₁, X₂, ..., X_n (with trivial exceptions) iff n≤(^k+m _m+1). For example four moments and four linear combinations under general conditions (specified later) determine the distribution of n=56 independent random variables, but not of 57.