Geodesic
Max-semistable laws in extremes of stationary random sequences
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 2, pp. 402-410

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In this paper we consider stationary sequences under the validity of an extension of Leadbetter's condition $D(u_n)$. For these sequences we prove that, if $\{k_n\}$ is a nondecreasing integer sequence satisfying $\lim_{n\to+\infty}k_{n+1}/k_n=r\ge 1$, then the limit law for the maximum of the first $k_n$ variables is a max-semistable law. This generalizes the corresponding result for sequences of independent identically distributed random variables of Grinevich [Theory Probab. Appl., 38 (1993), pp. 640–650] and the extremal types theorem of Leadbetter [Z. Wahrsch. Verw. Gebiete, 28 (1974), pp. 289–303]. We also prove that the limiting behavior of this maximum can be inferred from the limiting behavior of the corresponding maximum of the associated independent sequence, and we extend the well-known notion of extremal index. An illustrative example is given.
Keywords: weak convergence, stationarity, max-semistable laws.
Mots-clés : maximum 
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     author = {M. G. Temido and L. Canto E. Castro},
     title = {Max-semistable laws in extremes of stationary random sequences},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {402--410},
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     number = {2},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a18/}
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M. G. Temido; L. Canto E. Castro. Max-semistable laws in extremes of stationary random sequences. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 2, pp. 402-410. http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a18/