Abstract
Let {X n } be a stationary sequence and {k n } be a nondecreasing sequence such that k n+1/k n →r≥1. Assume that the limit distribution G of \(M_{k_{n}}\) with an appropriate linear normalization exists. We consider the maxima M n =max {X i ,i≤n} sampled at random times T n , where T n /k n converges in probability to a positive random variable D, and show that the limit distribution of \(M_{T_{n}}\) exists under weak mixing conditions. The limit distribution of \(M_{T_{n}}\) is a mixture of G and the distribution of D.
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References
Alpuim MT (1988) Contribuições à teoria de extremos em sucessões dependentes. PhD thesis, University of Lisbon
Alpuim MT, Catkan NA, Hüsler J (1995) Extremes and clustering of nonstationary max-AR(1) sequences. Stoch Process Appl 56:171–184
Anderson CW (1970) Extreme value theory for a class of discrete distribution with applications to some stochastic processes. J Appl Probab 7:99–113
Barndorff-Nielson O (1964) On a limit distribution of the maximum of a random number of independent random variables. Acta Math Acad Sci Hung 15:399–403
Becker-Kern P (2001) Max-semistable hemigroups: structure, domains of attraction and limit theorems with random sample size. Probab Math Stat 21:441–465
Canto e Castro L, de Haan L, Temido MG (2002) Rarely observed sample maxima. Theory Probab Appl 45:779–782
Chernick MR (1981) A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann Probab 9:145–149
Ferreira H (1995) Extremes of a random number of variables from periodic sequences. J Stat Plan Inference 45:133–141
Galambos J (1978) The theory of extreme of order statistics. Wiley, New York
Grinevich IV (1993) Domains of attraction of the max-semistable laws under linear and power normalizations. Theory Probab Appl 38:640–650
Hall A (2003) Extremes of integer-valued moving averages models with exponential type-tails. Extremes 6:361–379
Hall A, Temido MG (2007) On the maximum term of MA and max-AR models with margins in Anderson’s class. Theory Probab Appl 51:291–304
Hall A, Temido MG (2009) On the max-semistable limit of maxima of stationary sequences with missing values. J Stat Plan Inference 139:875–890
Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, Berlin
Pancheva E (1992) Multivariate max-semistable distributions. Theory Probab Appl 18:679–705
Rootzén H (1986) Extreme value theory for moving average processes. Ann Probab 14:612–652
Silvestrov DS, Teugels JL (1998) Limit theorems for extremes with random sample sizes. Adv Appl Probab 30:777–806
Temido MG, Canto e Castro L (2003) Max-semistable laws in extremes of stationary random sequences. Theory Probab Appl 47:365–374
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Freitas, A., Hüsler, J. & Temido, M.G. Limit laws for maxima of a stationary random sequence with random sample size. TEST 21, 116–131 (2012). https://doi.org/10.1007/s11749-011-0238-2
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DOI: https://doi.org/10.1007/s11749-011-0238-2