On two extensions of the canonical Feller\u2013Spitzer distribution

Vladimir Vladimirovich Vinogradov,Richard Bruce Paris

doi:10.1186/s40488-021-00113-4

Vladimir Vladimirovich Vinogradov, Richard Bruce Paris

Open Access

https://doi.org/10.1186/s40488-021-00113-4

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Abstract

We introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.

Highlights

The discovery of new infinitely divisible distributions with regularly varying tails is important for the development of distribution theory per se as well as for mathematical modelling, and applications in statistics and to decision theory
Definition 2 Consider a class of positive infinitely divisible distributions which is indexed by the real-valued parameter ρ > 1/2, has the following density of its Lévy measure νρ(2)({·}) on R1+: τρ (x) := 2ρ−1 ·
The following result is of a particular value providing an illustration of the usefulness of numerous relatively recent advances in the theory of infinitely divisible distributions, most of which were summarized in (Steutel and van Harn (2004), Chapter V)

Summary

Introduction

The discovery of new infinitely divisible distributions with regularly varying tails is important for the development of distribution theory per se as well as for mathematical modelling, and applications in statistics and to decision theory. The following result is of a particular value providing an illustration of the usefulness of numerous relatively recent advances in the theory of infinitely divisible distributions, most of which were summarized in (Steutel and van Harn (2004), Chapter V) To some extent, it can be regarded as a counterpart of Theorem 2 of “Properties of the class Xρ” section. The self-decomposability of the r.v. Yρ follows by combining the inequality (110) of “Appendix” section which yields that ∀ρ > 1/2, the canonical density kρ(x) is a decreasing function in R1+, with (Steutel and van Harn (2004), Theorem V.2.11). (iii) In view of (Steutel and van Harn (2004), Theorem V.2.17), the monotonic decay of pρ(x) follows from that of the canonical density kρ(x) combined with the fact that formula (47) yields that for ρ > 1/2, kρ (0+) = 1.

Since for θ

An integration by parts applied to the first integral then yields