Reconstructing the Distribution from Partial Sums of Samples

Abstract

Let us observe an infinite sequence $z_1 = r_1 + \varepsilon_1, z_2 = r_2 + \varepsilon_2, \cdots$ where $r_1, r_2,\cdots$ are the partial sums of independent and identically distributed random variables and the sequence of random variables $\varepsilon_k$ (the errors) is bounded by a function $f(k)$. Knowing the sequence $z_n$ we want to determine the distribution function of the summands. We will show that this problem cannot be solved in general even if $f(k)$ is constant.