# On the asymptotic form of convex hulls of Gaussian random fields

Open Mathematics (2014)

• Volume: 12, Issue: 5, page 711-720
• ISSN: 2391-5455

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## Abstract

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We consider a centered Gaussian random field X = X t : t ∈ T with values in a Banach space $𝔹$ 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 = convX 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.

## How to cite

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Youri Davydov, and Vygantas Paulauskas. "On the asymptotic form of convex hulls of Gaussian random fields." Open Mathematics 12.5 (2014): 711-720. <http://eudml.org/doc/269078>.

@article{YouriDavydov2014,
abstract = {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 = convX 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.},
author = {Youri Davydov, Vygantas Paulauskas},
journal = {Open Mathematics},
keywords = {Gaussian processes and fields; Convex hull; Limit behavior; convex hull; limit behavior},
language = {eng},
number = {5},
pages = {711-720},
title = {On the asymptotic form of convex hulls of Gaussian random fields},
url = {http://eudml.org/doc/269078},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Youri Davydov
AU - Vygantas Paulauskas
TI - On the asymptotic form of convex hulls of Gaussian random fields
JO - Open Mathematics
PY - 2014
VL - 12
IS - 5
SP - 711
EP - 720
AB - 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 = convX 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.
LA - eng
KW - Gaussian processes and fields; Convex hull; Limit behavior; convex hull; limit behavior
UR - http://eudml.org/doc/269078
ER -

## References

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1. [1] Aliprantis C.D., Border K.C., Infinite Dimensional Analysis, 3rd ed., Springer, Berlin, 2006
2. [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 Zbl0109.11803
3. [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 Zbl1227.60034
4. [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 Zbl1260.60052
5. [5] Fernique X., Régularité de processus gaussiens, Invent. Math., 1971, 12(4), 304–320 http://dx.doi.org/10.1007/BF01403310 Zbl0217.21104
6. [6] Goodman V., Characteristics of normal samples, Ann. Probab., 1988, 16(3), 1281–1290 http://dx.doi.org/10.1214/aop/1176991690 Zbl0712.60006
7. [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 Zbl0518.60021
8. [8] Ledoux M., Talagrand M., Probability in Banach Spaces, Classics Math., Springer, Berlin, 1991 Zbl0748.60004
9. [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 Zbl1188.82024
10. [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 Zbl0341.60024
11. [11] Schneider R., Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia Math. Appl., 44, Cambridge University Press, Cambridge, 1993 Zbl0798.52001
12. [12] Schneider R., Recent results on random polytopes, Boll. Unione Mat. Ital., 2008, 9(1), 17–39 Zbl1206.52011

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