Special moments

Abstract

In this article, we show that a linear combination $X$ of $n$ independent, unbiased Bernoulli random variables $\{X_k\}$ can match the first $2n$ moments of a random variable $Y$ which is uniform on an interval. More generally, for each $p \ge 2$, each $X_k$ can be uniform on an arithmetic progression of length $p$. All values of $X$ lie in the range of $Y$, and their ordering as real numbers coincides with dictionary order on the vector $(X_1,...,X_n)$. The construction involves the roots of truncated $q$-exponential series. It applies to a construction in numerical cubature using error-correcting codes [arXiv:math.NA/0402047]. For example, when $n=2$ and $p=2$, the values of $X$ are the 4-point Chebyshev quadrature formula.