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Central limit theorems for the number of records in discrete models

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Raul Gouet ,
F. Javier López  and
Gerardo Sanz
Raul Gouet*
Affiliation:
Universidad de Chile
F. Javier López*
Affiliation:
Universidad de Zaragoza
Gerardo Sanz*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Departamento de Ingeniería Matemática y Centro de Modelamiento Matemático, Universidad de Chile, UMI-CNRS-2807, Casilla 170-3, Correo 3, Santiago, Chile. Email address: rgouet@dim.uchile.cl
∗∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
∗∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
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Abstract

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Consider a sequence (Xn) of independent and identically distributed random variables taking nonnegative integer values, and call Xn a record if Xn> max{X1,…,Xn−1}. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.

Keywords

Extrememartingalerecordcentral limit theorem

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 781 - 800
Copyright
Copyright © Applied Probability Trust 2005 

References

Ahsanullah, M. (1995). Record Statistics. Nova Science Publishers, Commack, NY.Google Scholar
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley, New York.CrossRefGoogle Scholar
Bai, Z., Hwang, H. and Liang, W. (1998). Normal approximation of the number of records in geometrically distributed random variables. Random Structures Algorithms 13, 319–334.3.0.CO;2-Y>CrossRefGoogle Scholar
Deheuvels, P. (1974). Valeurs extrémales d' échantillons croissants d' une variable aléatoire réelle. Ann. Inst. H. Poincaré Sect. B 10, 89–114.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Heidelberg.CrossRefGoogle Scholar
Flajolet, P. and Sedgewick, R. (1995). Mellin transforms and asymptotics: finite differences and Rice's integrals. Theoret. Comput. Sci. 144, 101–124.Google Scholar
Gouet, R., López, F. J. and San Miguel, M. (2001). A martingale approach to strong convergence of the number of records. Adv. Appl. Prob. 33, 864–873.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Nevzorov, V. B. (2001). Records: Mathematical Theory (Transl. Math. Monogr. 194). American Mathematical Society, Providence, RI.Google Scholar
Prodinger, H. (1996). Combinatorics of geometrically distributed random variables: left-to-right maxima. Discrete Math. 153, 253–270.Google Scholar
Pugh, W. (1990). Skip lists: a probabilistic alternative to balanced trees. Commun. ACM 32, 668–676.Google Scholar
Rényi, A. (1962). Théorie des éléments saillants d' une suite d' observations. Ann. Fac. Sci. Univ. Clermont-Ferrand 8, 7–13.Google Scholar
Vervaat, W. (1973). Limit theorems for records from discrete distributions. Stoch. Process. Appl. 1, 317–334.Google Scholar