On the weak convergence of extremes in some Banach spaces

Matsak, I.

doi:10.1090/S0094-9000-05-00621-6

On the weak convergence of extremes in some Banach spaces

Author: I. K. Matsak
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 69 (2003).
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

Abstract | References | Similar Articles | Additional Information

Abstract: The weak convergence of random elements

$\begin{displaymath}U_n=b_n (Z_n -a_n \mathfrak{S}) \end{displaymath}$

is studied for Banach spaces with an unconditional basis, where $Z_n= \max _{1\leq k \leq n} X_k$ and

, $k\geq 1$ , are independent copies of a random element

1. B. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. of Math. (2) 44 (1943), 423–453 (French). MR 8655, 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. 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
4. Ī. K. Matsak, Weak convergence of extreme values of independent random elements in Banach spaces with an unconditional basis, Ukraïn. Mat. Zh. 48 (1996), no. 6, 805–812 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 6, 905–913 (1997). MR 1418157, https://doi.org/10.1007/BF02384175
5. Ī. K. Matsak, On integral functionals of extremal random functions, Teor. Ĭmovīr. Mat. Stat. 65 (2001), 110–120 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 65 (2002), 123–134. MR 1936135
6. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
7. Jean Geffroy, Contribution à la théorie des valeurs extrêmes, Publ. Inst. Statist. Univ. Paris 7 (1958), no. 3-4, 37–121 (French). MR 105168
8. Contributions to order statistics, Edited by Ahmed E. Sarhan and Bernard G. Greenberg, John Wiley & Sons, Inc., New York-London, 1962. MR 0169336
9. Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
10. J. Galambos and E. Seneta, Regularly varying sequences, Proc. Amer. Math. Soc. 41 (1973), 110–116. MR 323963, https://doi.org/10.1090/S0002-9939-1973-0323963-5
11. N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, “Nauka”, Moscow, 1985 (Russian). MR 787803
12. 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
13. James Pickands III, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), 881–889. MR 224231, https://doi.org/10.1214/aoms/1177698320
14. S. A. Chobanjan and V. I. Tarieladze, Gaussian characterizations of certain Banach spaces, J. Multivariate Anal. 7 (1977), no. 1, 183–203. MR 433572, https://doi.org/10.1016/0047-259X(77)90038-0
15. Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR 0254888