On concentrated probabilities

Wojciech Bartoszek

Annales Polonici Mathematici (1995)

  • Volume: 61, Issue: 1, page 25-38
  • ISSN: 0066-2216

Abstract

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Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence g n ∈ G such that μ ∗ n ( g n A ) ≡ 1 for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power μ ∗ k has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological group G.

How to cite

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Wojciech Bartoszek. "On concentrated probabilities." Annales Polonici Mathematici 61.1 (1995): 25-38. <http://eudml.org/doc/262419>.

@article{WojciechBartoszek1995,
abstract = {Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence $g_n ∈ G$ such that $μ^\{∗n\}(g_n A) ≡ 1$ for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power $μ^\{∗k\}$ has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological group G.},
author = {Wojciech Bartoszek},
journal = {Annales Polonici Mathematici},
keywords = {random walk; concentration function; convolution operator; locally compact group; probability measure; concentration functions; Haar measure; scattered},
language = {eng},
number = {1},
pages = {25-38},
title = {On concentrated probabilities},
url = {http://eudml.org/doc/262419},
volume = {61},
year = {1995},
}

TY - JOUR
AU - Wojciech Bartoszek
TI - On concentrated probabilities
JO - Annales Polonici Mathematici
PY - 1995
VL - 61
IS - 1
SP - 25
EP - 38
AB - Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence $g_n ∈ G$ such that $μ^{∗n}(g_n A) ≡ 1$ for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power $μ^{∗k}$ has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological group G.
LA - eng
KW - random walk; concentration function; convolution operator; locally compact group; probability measure; concentration functions; Haar measure; scattered
UR - http://eudml.org/doc/262419
ER -

References

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  1. [B1] W. Bartoszek, On the asymptotic behaviour of iterates of positive linear operators, Notices South African Math. Soc. 25 (1993), 48-78. 
  2. [B2] W. Bartoszek, On concentration functions on discrete groups, Ann. Probab., to appear. 
  3. [B3] W. Bartoszek, On the equation μ̌ ∗ϱ∗μ = ϱ, Demonstratio Math., to appear. 
  4. [C] I. Csiszár, On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups, Z. Wahrsch. Verw. Gebiete 5 (1966), 279-299. Zbl0144.39504
  5. [CFS] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, New York, 1981. 
  6. [DL1] Y. Derriennic et M. Lin, Sur le comportement asymptotique de puissances de convolution d'une probabilité, Ann. Inst. H. Poincaré 20 (1984), 127-132. Zbl0536.60014
  7. [DL2] Y. Derriennic et M. Lin, Convergence of iterates of averages of certain operator representations and of convolution powers, J. Funct. Anal. 85 (1989), 86-102. Zbl0712.22008
  8. [H] H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin, 1977. Zbl0376.60002
  9. [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1963. 
  10. [HM] K. H. Hofmann and A. Mukherjea, Concentration functions and a class of non-compact groups, Math. Ann. 256 (1981), 535-548. Zbl0471.60015
  11. [M] A. Mukherjea, Limit theorems for probability measures on non-compact groups and semigroups, Z. Wahrsch. Verw. Gebiete 33 (1976), 273-284. Zbl0304.60004
  12. [P] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. Zbl0153.19101
  13. [T] A. Tortrat, Lois de probabilité sur un espace topologique complétement régulier et produits infinis à termes indépendants dans un groupe topologique, Ann. Inst. H. Poincaré Sect. B 1 (1965), 217-237 Zbl0137.35301

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