Measure convolution semigroups and non-infinitely divisible probability distributions

Abstract

Let $μ$ be a probability measure (or corresponding random variable) such that all moments $μ_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible, and in particular lognormal, distributions lose infinitely divisibilty when censored in certain ways even if all moments are arbitrarily close to those of the uncensored distribution. The moments of a composition of k copies of $μ$ are expressed as combinatorial compositions of the $μ_n$. We express the moments of the compositions in the context of occupancy problems, arranging n balls in k cells; the classical convolution is described by Maxwell-Boltzmann statistics and is multinomial. For certain non-infinitely divisible measures with moments increasing fast enough the indexing of a k-cell combinatorial composition is extended to indexing by non-negative real t and we construct classical convolution measure semigroups from amongst the t-indexed classes. We prove also that when a random variable with infinitely divisible distribution is embedded in a Lévy process $(Y_t)$ then the t-indexed Maxwell-Boltzmann is the law of $Y_t$. In order to get moment-based multinomial compositions indexed by a continuum we use random measures and random distributions rather than random variables. An alternative approach to embeddability of a non-infinitely divisible $μ$ is by considering non-classical convolution measure semigroups; for example embedding $μ$ in a Boolean convolution measure semigroup and retaining the multinomial character of the moments.embedding $μ$ in a Boolean convolution measure semigroup and retaining the multinomial character of the moments. Embedding $μ$ in an Urbanik convolution measure semigroup loses their multinomial character .