Limiting behaviour of the stationary search cost distribution driven by a generalized gamma process

TL;DR

This paper analyzes the limiting behavior of the stationary search cost distribution in a move-to-front Markov chain with infinitely many objects, where probabilities are derived from a generalized gamma process, providing exact moments and extensions.

Contribution

It introduces an exact formula for moments of the stationary search cost distribution when probabilities are based on a generalized gamma subordinator, extending to the Pitman-Yor process.

Findings

Derived exact moments for the stationary search cost distribution.

Applied results to generalized gamma and Pitman-Yor processes.

Provided insights into the asymptotic behavior of search costs.

Abstract

Consider a list of labeled objects that are organized in a heap. At each time, object $j$ is selected with probability $p_j$ and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities $p_j$'s are defined as the normalization of the increments of a subordinator. In this setting, we provide an exact formula for the moments of any order of the stationary search cost distribution. We illustrate the new findings in the case of a generalized gamma subordinator and deal with an extension to the two--parameter Poisson--Dirichlet process, also known as Pitman--Yor process.