IDENTIFIABILITY OF DISTRIBUTIONS OF INDEPENDENT RANDOM VARIABLES BY LINEAR COMBINATIONS AND MOMENTS

Let X 1 , X 2 , ..., 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=1 n a ij X j , i=1, 2,...,k with an arbitrary nonvanishing joint characteristic function uniquely determines the distributions of X 1 , X 2 , ..., 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.