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2014 | 12 | 5 | 711-720
Tytuł artykułu

On the asymptotic form of convex hulls of Gaussian random fields

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider a centered Gaussian random field X = {X t : t ∈ T} with values in a Banach space $$\mathbb{B}$$ defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = conv{X t : t ∈ T n}, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W n), where f is some function, is also studied.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
5
Strony
711-720
Opis fizyczny
Daty
wydano
2014-05-01
online
2014-02-15
Bibliografia
  • [1] Aliprantis C.D., Border K.C., Infinite Dimensional Analysis, 3rd ed., Springer, Berlin, 2006
  • [2] Berman S.M., A law of large numbers for the maximum in a stationary Gaussian sequence, Ann. Math. Statist., 1962, 33, 93–97 http://dx.doi.org/10.1214/aoms/1177704714
  • [3] Davydov Y., On convex hull of Gaussian samples, Lith. Math. J., 2011, 51(2), 171–179 http://dx.doi.org/10.1007/s10986-011-9117-5
  • [4] Davydov Y., Dombry C., Asymptotic behavior of the convex hull of a stationary Gaussian process, Lith. Math. J., 2012, 52(4), 363–368 http://dx.doi.org/10.1007/s10986-012-9179-z
  • [5] Fernique X., Régularité de processus gaussiens, Invent. Math., 1971, 12(4), 304–320 http://dx.doi.org/10.1007/BF01403310
  • [6] Goodman V., Characteristics of normal samples, Ann. Probab., 1988, 16(3), 1281–1290 http://dx.doi.org/10.1214/aop/1176991690
  • [7] Leadbetter M.R., Lindgren G., Rootzén H., Extremes and Related Properties of Random Sequences and Processes, Springer Ser. Statist., Springer, New York-Berlin, 1983 http://dx.doi.org/10.1007/978-1-4612-5449-2
  • [8] Ledoux M., Talagrand M., Probability in Banach Spaces, Classics Math., Springer, Berlin, 1991
  • [9] Majumdar S.N., Comtet A., Randon-Furling J., Random convex hulls and extreme value statistics, J. Stat. Phys., 2010, 138(6), 955–1009 http://dx.doi.org/10.1007/s10955-009-9905-z
  • [10] Mittal Y., Ylvisaker D., Strong law for the maxima of stationary Gaussian processes, Ann. Probab., 1976, 4(3), 357–371 http://dx.doi.org/10.1214/aop/1176996085
  • [11] Schneider R., Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia Math. Appl., 44, Cambridge University Press, Cambridge, 1993
  • [12] Schneider R., Recent results on random polytopes, Boll. Unione Mat. Ital., 2008, 9(1), 17–39
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0375-9
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