A counting process in the max-scheme
Author:
I. K. Matsak
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 91 (2015), 115-129
MSC (2010):
Primary 60G70
DOI:
https://doi.org/10.1090/tpms/971
Published electronically:
February 4, 2016
MathSciNet review:
3364128
Full-text PDF Free Access
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Additional Information
Abstract: The exact asymptotic behavior of a counting process is studied for the max-scheme in the case of independent random variables.
References
- V. V. Buldygin, K.-H. Indlekofer, O. I. Klesov, and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, ‘TViMS”, Kyiv, 2012. (Ukrainian)
- 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
- 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
- James Pickands III, An iterated logarithm law for the maximum in a stationary Gaussian sequence, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 12 (1969), 344–353. MR 251776, DOI https://doi.org/10.1007/BF00538755
- Laurens de Haan and Arie Hordijk, The rate of growth of sample maxima, Ann. Math. Statist. 43 (1972), 1185–1196. MR 312550, DOI https://doi.org/10.1214/aoms/1177692470
- Michael J. Klass, The Robbins-Siegmund series criterion for partial maxima, Ann. Probab. 13 (1985), no. 4, 1369–1370. MR 806233
- Laurens de Haan and Ana Ferreira, Extreme value theory, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. An introduction. MR 2234156
- K. S. Akbash and I. K. Matsak, One improvement of the law of the iterated logarithm for the maximum scheme, Ukrainian Math. J. 64 (2013), no. 8, 1290–1296. MR 3104866, DOI https://doi.org/10.1007/s11253-013-0716-7
- I. K. Matsak, Asymptotic behavior of a counting process in the maximum scheme, Ukrainian Math. J. 65 (2014), no. 11, 1743–1748. Translation of Ukraïn. Mat. Zh. 65 (2013), no. 11, 1575–1579. MR 3216868, DOI https://doi.org/10.1007/s11253-014-0893-z
- I. Matsak and I. Rozora, Asymptotic behaviour of counting process in the max-scheme. Discrete case, Georg. Math. J. (to appear)
- G. M. Fikhtengol’ts, Course of Differential and Integral Calculus, vol. 2, “Nauka”, Moscow, 1969. (Russian)
- M. R. Leadbetter, Georg Lindgren, and Holger Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, Springer-Verlag, New York-Berlin, 1983. MR 691492
- Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
- G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. I: Series, integral calculus, theory of functions, Springer-Verlag, New York-Berlin, 1972. Translated from the German by D. Aeppli; Die Grundlehren der mathematischen Wissenschaften, Band 193. MR 0344042
References
- V. V. Buldygin, K.-H. Indlekofer, O. I. Klesov, and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, ‘TViMS”, Kyiv, 2012. (Ukrainian)
- J. Galambos, The Asymptotic Theory of Extreme Order Statistics, John Wiley & Sons, New York, 1978. MR 489334 (80b:60040)
- Ole Barndorff-Nielsen, On the limit behaviour of extreme order statistics, Ann. Math. Statist. 34 (1963), no. 3, 992–1002. MR 0150889 (27:875)
- J. Pickands, An iterated logarithm law for the maximum in a stationary Gaussian sequence, Z. Warhscheinlichkeitstheorie verw. Geb. 12 (1969), no. 3, 344–355. MR 0251776 (40:5003)
- L. de Haan and A. Hordijk, The rate of growth of sample maxima, Ann. Math. Statist. 43 (1972), 1185–1196. MR 0312550 (47:1107)
- M. J. Klass, The Robbins–Siegmund criterion for partial maxima, Ann. Probab. 13 (1985), 1369–1370. MR 806233 (87b:60046)
- L. de Haan and A. Ferreira, Extreme Values Theory: An Introduction, Springer, Berlin, 2006. MR 2234156 (2007g:62008)
- K. S. Akbash and I. K. Matsak, An improvement of the law of the iterated logarithm for the max-scheme, Ukr. Mat. Zh. 64 (2012), no. 8, 1132–1137; English transl in Ukrainian Math. J. 64 (2013), no. 8, 1290–1296. MR 3104866
- I. K. Matsak, Asymptotic behavior of a counting process in the max-scheme, Ukr. Mat. Zh. 65 (2013), no. 11, 1575–1579; English transl in Ukrainian Math. J. 65 (2014), no. 11, 1743–1748. MR 3216868
- I. Matsak and I. Rozora, Asymptotic behaviour of counting process in the max-scheme. Discrete case, Georg. Math. J. (to appear)
- G. M. Fikhtengol’ts, Course of Differential and Integral Calculus, vol. 2, “Nauka”, Moscow, 1969. (Russian)
- M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York, Heidelberg, Berlin, 1983. MR 691492 (84h:60050)
- E. Seneta, Regularly Varying Functions, Springer-Verlag, New York–Heidelberg–Berlin, 1976. MR 0453936 (56:12189)
- G. Pólya and G. Szegö, Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions, Springer-Verlag, New York–Heidelberg–Berlin, 1972. MR 0344042 (49:8782)
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Additional Information
I. K. Matsak
Affiliation:
Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
ivanmatsak@univ.kiev.ua
Keywords:
Maximum of independent random variables,
counting process,
almost sure behavior
Received by editor(s):
August 6, 2014
Published electronically:
February 4, 2016
Article copyright:
© Copyright 2016
American Mathematical Society
