On the weak convergence of extremes in some Banach spaces
Author:
I. K. Matsak
Translated by:
V. Semenov
Journal:
Theor. Probability and Math. Statist. 69 (2004), 141-152
MSC (2000):
Primary 60B12
DOI:
https://doi.org/10.1090/S0094-9000-05-00621-6
Published electronically:
February 9, 2005
MathSciNet review:
2110912
Full-text PDF Free Access
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Abstract: The weak convergence of random elements \[ U_n=b_n (Z_n -a_n \mathfrak {S}) \] is studied for Banach spaces with an unconditional basis, where $Z_n= \max _{1\leq k \leq n} X_k$ and $X_k$, $k\geq 1$, are independent copies of a random element $X$.
References
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References
- B. V. Gnedenko, Sur la distribution limit du terme maximum d’une série aléatoire, Ann. Math. 44 (1943), 423–453. MR 0008655 (5:41b)
- J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York–Chichester–Brisbane–Toronto, 1978. MR 0489334 (80b:60040)
- M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer, New York, 1983. MR 0691492 (84h:60050)
- I. K. Matsak, The weak convergence of extreme values of independent random elements in Banach spaces with an unconditional basis, Ukr. Matem. Zh. 48 (1996), no. 6, 805–812; English. transl. in Ukrain. Math. J. 48 (1996), no. 6, 905–913 (1997). MR 1418157 (97i:60007)
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- I. K. Matsak and A. M. Plichko, On maxima of independent random elements in a Banach function lattice, Teor. Imovir. Matem. Statist. 61 (1999), 105–116; English transl. in Theory Probab. Math. Statist. 61 (2000), 109–120. MR 1866964 (2002k:60020)
- J. Pickands, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), 881–889. MR 0224231 (36:7275)
- S. A. Chobanyan and V. I. Tarieladze, Gaussian characterizations of certain Banach spaces, J. Multivar. Anal. 7 (1977), no. 1, 183–203. MR 0433572 (55:6545)
- J.-P. Kahane, Some Random Series of Functions, D. C. Heath, Lexington, MA, 1968. MR 0254888 (40:8095)
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Additional Information
I. K. Matsak
Affiliation:
Kyiv National University of Technology and Design, Nemyrovych-Danchenko Street 2, Kyiv 02011, Ukraine
Email:
infor1@vtv.kiev.ua
Received by editor(s):
June 26, 2002
Published electronically:
February 9, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
