# On concentrated probabilities

Annales Polonici Mathematici (1995)

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

## Access Full Article

top## Abstract

top## How to cite

topWojciech 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

top- [B1] W. Bartoszek, On the asymptotic behaviour of iterates of positive linear operators, Notices South African Math. Soc. 25 (1993), 48-78.
- [B2] W. Bartoszek, On concentration functions on discrete groups, Ann. Probab., to appear.
- [B3] W. Bartoszek, On the equation μ̌ ∗ϱ∗μ = ϱ, Demonstratio Math., to appear.
- [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
- [CFS] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, New York, 1981.
- [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
- [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
- [H] H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin, 1977. Zbl0376.60002
- [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1963.
- [HM] K. H. Hofmann and A. Mukherjea, Concentration functions and a class of non-compact groups, Math. Ann. 256 (1981), 535-548. Zbl0471.60015
- [M] A. Mukherjea, Limit theorems for probability measures on non-compact groups and semigroups, Z. Wahrsch. Verw. Gebiete 33 (1976), 273-284. Zbl0304.60004
- [P] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. Zbl0153.19101
- [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

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.