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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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1519354834

Digital Object Identifier
doi:10.1214/18-ECP111

Mathematical Reviews number (MathSciNet)
MR3771769

Zentralblatt MATH identifier
1390.60181

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

  • [1] Barrera, J. and Fontbona, J.: The limiting Move-to-Front search cost in law of large numbers asymptotic regimes. The Ann. Appl. Probab., 20, (2010), 722–752.
  • [2] Barrera, J. and Huillet, T. and Paroissin, C.: Size-biased permutation of Dirichlet partitions and search-cost distribution. Probab. Engrg. Inform. Sci., 19, (2005), 83–97.
  • [3] Barrera, J. and Huillet, T. and Paroissin, C.. Limiting search cost distribution for the move-to-front rule with random request probabilities. Operat. Res. Letters, 34, (2006), 557–563.
  • [4] Barrera, J. and Paroissin, C.: On the distribution of the stationary search cost for the move-to-front rule with random weights. J. Appl. Probab., 41, (2004), 250–262.
  • [5] Billingsley, P.: Probability and Measure. Academic Press, 1995.
  • [6] Brix, A.: Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Probab., 31, (1999), 929–953.
  • [7] Caron, F. and Teh, Y. and Murphy, T. B.: Bayesian nonparametric Plackett-Luce models for the analysis of preferences for college degree programmes. Ann. Appl. Stat., 8, (2014), 1145–1181.
  • [8] Donnelly, P.: The heaps process, libraries, and size-biased permutations. J. Appl. Probab., 28, (1991), 321–335.
  • [9] Fill, J. A.: Limits and rate of convergence for the distribution of search cost under the move-to-front rule. Theoret. Comput. Sci., 164, (1996), 185–206.
  • [10] Fill, J. A.: An exact formula for the move-to-front rule for self-organizing lists.J. Theoret. Probab., 9, (1996), 113–160.
  • [11] Fill, J. A. and Holst, L.: On the distribution of search cost for the move-to-front rule. Random Structures Algorithms, 8, (1996), 179–186.
  • [12] Gradshteyn, I. S. and Ryzhik, I. M.: Tables of integrals, series and products. Academic Press, 2007.
  • [13] Hattori, K. and Hattori, T.: Sales ranks, burgers-like equations, and least-recently-used caching. Dedicated to Prof. K. R. Ito on his 60th birthday, (2010), B21:149–161.
  • [14] Jelenković, P. R.: Asymptotic approximation of the move-to-front search cost distribution and least-recently used caching fault probabilities. Ann. Appl. Probab., 9, (1999), 430–464.
  • [15] Jelenković, P. R. and Radovanović, A.: The persistent-access-caching algorithm. Random Structures Algorithms, 33, (2008), 219–251.
  • [16] Kingman, J. F. C.: Random discrete distributions. J. Roy. Stat. Soc., Series B, 37, (1975), 1–22.
  • [17] Kingman, J. F. C.: Poisson processes. Oxford University Press, Oxford, 1993.
  • [18] Leisen, F. and Lijoi, A. and Paroissin, C.: Limiting Behaviour of the search cost distribution for the move to front rule in the stable case. Statistics and Probability Letters, 81, (2011), 1827–1832.
  • [19] Lijoi, A. and Mena, R. H. and Prünster, I.: Controlling the reinforcement in Bayesian non–parametric mixture models.J. Roy. Statist. Soc., Ser. B, 69, (2007), 715–740.
  • [20] Lijoi, A. and Prünster, I.: A note on the problem of heaps. Sankhyā, 66, (2004), 232–240.
  • [21] Luce, R. D.: Individual Choice Behavior: A Theoretical Analysis. Wiley, New York. 1959.
  • [22] McCabe, J.: On serial files with relocatable records. Oper. Res., 13, (1965), 609–618.
  • [23] Plackett, R. L.: The Analysis of Permutations. Journal of the Royal Statistical Society-Series C, 24, (1975), 193–202.
  • [24] Perman, M. and Pitman, J. and Yor, M.: Size-biased sampling of Poisson point processes and excursions. Probability Theory and Related Fields, 92, (1992), 21–39.
  • [25] Pitman, J.: Combinatorial Stochastic Processes. Springer, Leture notes in Mathematics, New York. 2006.
  • [26] Pitman, J. and Yor, M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., 25, (1997), 855–900.
  • [27] Pitman, J. and Tran, N. M.: Size-biased permutation of a finite sequence with independent and identically distributed terms. Bernoulli, 21, (2015), 2484–2512.
  • [28] Regazzini, E. and Lijoi, A. and Prünster, I.: Distributional results for means of normalized random measures of independent increments. Ann. Statist., 31, (2003), 560–585.
  • [29] Tsetlin, M. L.: Finite automata and models of simple forms of behavior. Russian Math. Surveys, 18, (1963), 1–27.