Stieltjes classes for moment-indeterminate probability distributions

Abstract

LetFbe a probability distribution function with densityf. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments forFhas a nonunique solution (Fis M-indeterminate). Our goal is to describe a, wherehis a ‘small' perturbation function. Such a classSconsists of different distributionsFε(fεis the density ofFε) all sharing the same moments as those ofF, thus illustrating the nonuniqueness ofF, and of anyFε,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 ofScalled anindex of dissimilarityand 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.