Asymptotic stability of the maximum of normal stochastic processes
Author:
I. K. Matsak
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 79 (2009), 101-106
MSC (2000):
Primary 60B12
DOI:
https://doi.org/10.1090/S0094-9000-09-00784-4
Published electronically:
December 30, 2009
MathSciNet review:
2494539
Full-text PDF Free Access
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Additional Information
Abstract: Under quite general conditions, we prove that the maximum of a sequence of normal stochastic processes in the space $C_{[0,1]}$ is asymptotically stable almost surely.
References
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- Ole Barndorff-Nielsen, On the limit behaviour of extreme order statistics, Ann. Math. Statist. 34 (1963), 992–1002. MR 150889, DOI https://doi.org/10.1214/aoms/1177704022
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- Ī. K. Matsak and A. M. Plīchko, On the maxima of independent random elements in a Banach functional lattice, Teor. Ĭmovīr. Mat. Stat. 61 (1999), 105–116 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 61 (2000), 109–120 (2001). MR 1866964
- Ī. K. Matsak and A. M. Plīchko, Limit theorems for random elements in ideals of ordered bounded elements of functional Banach lattices, Ukraïn. Mat. Zh. 53 (2001), no. 1, 41–49 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 53 (2001), no. 1, 48–58. MR 1834638, DOI https://doi.org/10.1023/A%3A1010484700083
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- R. M. Dudley, Sample functions of the Gaussian process, Ann. Probability 1 (1973), no. 1, 66–103. MR 346884, DOI https://doi.org/10.1214/aop/1176997026
- Ī. K. Matsak, On limit points of a sequence of extreme values of normal random elements of a Banach space with an unconditional basis, Teor. Ĭmovīr. Mat. Stat. 55 (1996), 136–143 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 55 (1997), 143–151 (1998). MR 1641569
- X. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, École d’Été de Probabilités de Saint-Flour, IV-1974, Springer, Berlin, 1975, pp. 1–96. Lecture Notes in Math., Vol. 480 (French). MR 0413238
References
- B. V. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. of Math. (2) 44 (1943), no. 3, 423–453. MR 0008655 (5:41b)
- J. Galambos, The Asymptotic Theory of Extreme Order Statistics, John Wiley & Sons, New York–Chichester–Brisbane, 1978. MR 489334 (80b:60040)
- O. Barndorff-Nielsen, On the limit behaviour of extreme order statistics, Ann. Math. Statist. 34 (1963), no. 3, 992–1002. MR 0150889 (27:875)
- I. K. Matsak, A limit theorem for the maximum of Gaussian independent random variables in the space $C$, Ukrain. Mat. Zh. 47 (1995), no. 7, 1006–1008; English transl. in Ukrainian Math. J. 47 (1995), no. 7, 1152–1155. MR 1367958
- I. K. Matsak and A. M. Plichko, On the maxima of independent random elements in a Banach functional lattice, Teor. Imovir. Mat. Stat. 61 (1999), 105–116; English transl. in Theory Probab. Math. Statist. 61 (2000), 109–120. MR 1866964 (2002k:60020)
- I. K. Matsak and A. M. Plichko, Limit theorems for random elements in ideals of ordered bounded elements of functional Banach lattices, Ukrain. Mat. Zh. 53 (2001), no. 1, 41–49; English transl. in Ukrainian Math. J. 53 (2001), no. 1, 48–58. MR 1834638 (2002d:60002)
- I. K. Matsak, On the relative stability of extremal random functions, Mat. Zametki 71 (2002), no. 5, 787–790; English transl. in Math. Notes 71 (2002), no. 5, 717–720. MR 1936843 (2003k:60114)
- R. M. Dudley, Sample functions of the Gaussian process, Ann. Probab. 1 (1973), no. 1, 66–103. MR 0346884 (49:11605)
- I. K. Matsak, On limit points of a sequence of extreme values of normal random elements of a Banach space with an unconditional basis, Teor. Imovir. Mat. Stat. 55 (1996), 136–143; English transl. in Theory Probab. Math. Statist. 55 (1997), 143–151. MR 1641569 (99g:60011)
- X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, Lecture Notes in Mathematics, Springer-Verlag, Berlin–Heidelberg, 1975. MR 0413238 (54:1355)
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Additional Information
I. K. Matsak
Affiliation:
Department of Operation Research, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
mik@unicyb.kiev.ua
Keywords:
Asymptotic stability,
extremal values,
normal stochastic processes,
the space $C_{[0,1]}$
Received by editor(s):
January 30, 2007
Published electronically:
December 30, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
