Does the ratio of Laplace transforms of powers of a function identify the function?

Abstract

In auction theory, one is interested in identifying the distribution of bids based on the distribution of the highest ones. We study this problem as a special case of the following question. Let m, n be two distinct nonnegative integers and f a nonzero measurable function on [0, ∞) of at most exponential order. Let H n,m ∶= f n f m be the ratio of the Laplace transforms of f n and f m . Does knowledge of the function H n,m uniquely specify the function f ? This is a generalization of Lerch's theorem (Laplace transform specifies the function). Under some rather strong assumptions on f we show that the answer is affirmative.