Stieltjes classes for moment-indeterminate probability distributions

Abstract

Let F be a probability distribution function with density f. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments for F has a nonunique solution (F is M-indeterminate). Our goal is to describe a Stieltjes class S = {f_ε = f[1 + εh], ε ∈ [-1, 1]}, where h is a `small' perturbation function. Such a class S consists of different distributions F_ε (f_ε is the density of F_ε) all sharing the same moments as those of F, thus illustrating the nonuniqueness of F, and of any F_ε, in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic of S called an index of dissimilarity and calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.