Electronic Communications in Probability

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

Alfred Kume, Fabrizio Leisen, and Antonio Lijoiï

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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.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 11, 10 pp.

Received: 11 April 2017
Accepted: 25 January 2018
First available in Project Euclid: 23 February 2018

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Zentralblatt MATH identifier

Primary: 60G57: Random measures
Secondary: 60G51: Processes with independent increments; Lévy processes

$\gamma $–stable process generalized gamma process heaps move-to-front rule, search cost distribution subordinator two–parameter Poisson–Dirichlet process

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Kume, Alfred; Leisen, Fabrizio; Lijoiï, Antonio. Limiting behaviour of the stationary search cost distribution driven by a generalized gamma process. Electron. Commun. Probab. 23 (2018), paper no. 11, 10 pp. doi:10.1214/18-ECP111. https://projecteuclid.org/euclid.ecp/1519354834

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