Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces

G. Krupa

Studia Mathematica (1998)

  • Volume: 131, Issue: 3, page 289-302
  • ISSN: 0039-3223

Abstract

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Under different compactness assumptions pointwise and mean ergodic theorems for subadditive superstationary families of random sets whose values are weakly (or strongly) compact convex subsets of a separable Banach space are presented. The results generalize those of [14], where random sets in ℝ d are considered. Techniques used here are inspired by [3].

How to cite

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Krupa, G.. "Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces." Studia Mathematica 131.3 (1998): 289-302. <http://eudml.org/doc/216581>.

@article{Krupa1998,
abstract = {Under different compactness assumptions pointwise and mean ergodic theorems for subadditive superstationary families of random sets whose values are weakly (or strongly) compact convex subsets of a separable Banach space are presented. The results generalize those of [14], where random sets in $ℝ^d$ are considered. Techniques used here are inspired by [3].},
author = {Krupa, G.},
journal = {Studia Mathematica},
keywords = {multivalued ergodic theorems; measurable multifunctions; random sets; subadditive superstationary processes; set convergence; pointwise and mean ergodic theorems; subadditive superstationary families; random subsets; separable Banach space},
language = {eng},
number = {3},
pages = {289-302},
title = {Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces},
url = {http://eudml.org/doc/216581},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Krupa, G.
TI - Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 3
SP - 289
EP - 302
AB - Under different compactness assumptions pointwise and mean ergodic theorems for subadditive superstationary families of random sets whose values are weakly (or strongly) compact convex subsets of a separable Banach space are presented. The results generalize those of [14], where random sets in $ℝ^d$ are considered. Techniques used here are inspired by [3].
LA - eng
KW - multivalued ergodic theorems; measurable multifunctions; random sets; subadditive superstationary processes; set convergence; pointwise and mean ergodic theorems; subadditive superstationary families; random subsets; separable Banach space
UR - http://eudml.org/doc/216581
ER -

References

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  2. [2] Z. Artstein and J. C. Hansen, Convexification in limit laws of random sets in Banach spaces, Ann. Probab. 13 (1985), 307-309. Zbl0554.60022
  3. [3] E. J. Balder and Ch. Hess, Two generalizations of Komlós' theorem with lower closure-type applications, J. Convex Anal. 3 (1996), 25-44. 
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  8. [8] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1979. 
  9. [9] J. M. Hammersley, Postulates for subadditive processes, Ann. Probab. 2 (1974), 652-680. Zbl0303.60044
  10. [10] Ch. Hess, On multivalued martingales whose values may be unbounded: Martingale selectors and Mosco convergence, J. Multivariate Anal. 39 (1991), 175-201. Zbl0746.60051
  11. [11] F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, ibid. 7 (1977), 149-182. Zbl0368.60006
  12. [12] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968), 499-510. Zbl0182.22802
  13. [13] U. Krengel, Un théorème ergodique pour les processus sur-stationnaires, C. R. Acad. Sci. Paris Sér. A 282 (1976), 1019-1021. Zbl0343.60017
  14. [14] K. Schürger, Ergodic theorems for subadditive superstationary families of convex compact random sets, Z. Wahrsch. Verw. Gebiete 62 (1983), 125-135. Zbl0489.60005
  15. [15] F. A. Valentine, Convex Sets, McGraw-Hill and Wiley, New York, 1968. 

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