Reversing conditional orderings

We analyze some specific aspects concerning conditional orderings and relations among them. To this purpose we define a suitable concept of reversed conditional ordering and prove some related results. In particular we aim to compare the univariate stochastic orderings ≤ st, ≤ hr, and ≤ lr in terms of differences among different notions of conditional orderings. Some applications of our result to the analysis of positive dependence will be detailed. We concentrate attention to the case of a pair of scalar random variables X, Y ​. Suitable extensions to multivariate cases are possible

On the concavity of the infinitesimal renewal function

The expected time of the first passage above a level of a process with stationary independent non-negative increments is studied. It is called infinitesimal renewal function. Results analogous to Brown's (1980) concavity result on the conventional renewal function, for the infinitesimal renewal function, are proved. Some connections with Brown's (1981) conjecture and shock models are pointed out.Infinitesimal renewal function DFR distribution geometric compounding Poisson shock model

A note on preservation of self-decomposability under geometric compounding

In this note Kolmogorov's canonical representations for geometric compounds of i.i.d. random variables are computed. Using this the geometric compounds of some self-decomposable (L-class) distributions are investigated. It is proved that the L-class is not closed under geometric compounding. Some nontrivial examples of self-decomposable distributions are given. Implications in queueing theory are pointed out.L-class geometric compounding Kolmogorov's formula GI/G/1 queue