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Asymptotic stability of the maximum of normal stochastic processes


Author: I. K. Matsak
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
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
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Abstract | References | Similar Articles | 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.


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  • 1. B. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. of Math. (2) 44 (1943), 423–453 (French). MR 0008655, https://doi.org/10.2307/1968974
  • 2. Janos Galambos, The asymptotic theory of extreme order statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1978. Wiley Series in Probability and Mathematical Statistics. MR 489334
  • 3. Ole Barndorff-Nielsen, On the limit behaviour of extreme order statistics, Ann. Math. Statist. 34 (1963), 992–1002. MR 0150889, https://doi.org/10.1214/aoms/1177704022
  • 4. I. K. Matsak, A limit theorem for the maximum of Gaussian independent random variables in the space 𝐶, Ukraïn. Mat. Zh. 47 (1995), no. 7, 1006–1008 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 47 (1995), no. 7, 1152–1155 (1996). MR 1367958, https://doi.org/10.1007/BF01084912
  • 5. 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
  • 6. 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, https://doi.org/10.1023/A:1010484700083
  • 7. I. K. Matsak, On the relative stability of extremal random functions, Mat. Zametki 71 (2002), no. 5, 787–790 (Russian); English transl., Math. Notes 71 (2002), no. 5-6, 717–720. MR 1936843, https://doi.org/10.1023/A:1015800324311
  • 8. R. M. Dudley, Sample functions of the Gaussian process, Ann. Probability 1 (1973), no. 1, 66–103. MR 0346884
  • 9. 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
  • 10. 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

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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

DOI: https://doi.org/10.1090/S0094-9000-09-00784-4
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