CONVOLUTION OF BETA PRIME DISTRIBUTION

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for F-3(2)(1). Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turan's inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill's ratio is discussed in detail.

Automated summary

In this article, we establish some identities for the convolution of a beta prime distribution with itself, and provide proofs for these identities. By applying these identities, we can derive complete monotonicity properties for quotients of confluent hypergeometric functions. Furthermore, we also present a simple proof for Turan's inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind using a probability approach, with a detailed discussion on the case of Mill's ratio.