On concentrated probabilities
Wojciech Bartoszek
Annales Polonici Mathematici, Tome 62 (1995), p. 25-38 / Harvested from The Polish Digital Mathematics Library
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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 gn∈G such that μ∗n(gnA)≡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.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:262419 
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Wojciech Bartoszek. On concentrated probabilities. Annales Polonici Mathematici, Tome 62 (1995) pp. 25-38. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv61z1p25bwm/

[000] [B1] W. Bartoszek, On the asymptotic behaviour of iterates of positive linear operators, Notices South African Math. Soc. 25 (1993), 48-78.

[001] [B2] W. Bartoszek, On concentration functions on discrete groups, Ann. Probab., to appear.

[002] [B3] W. Bartoszek, On the equation μ̌ ∗ϱ∗μ = ϱ, Demonstratio Math., to appear.

[003] [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. | Zbl 0144.39504

[004] [CFS] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, New York, 1981.

[005] [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. | Zbl 0536.60014

[006] [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. | Zbl 0712.22008

[007] [H] H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin, 1977. | Zbl 0376.60002

[008] [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1963.

[009] [HM] K. H. Hofmann and A. Mukherjea, Concentration functions and a class of non-compact groups, Math. Ann. 256 (1981), 535-548. | Zbl 0471.60015

[010] [M] A. Mukherjea, Limit theorems for probability measures on non-compact groups and semigroups, Z. Wahrsch. Verw. Gebiete 33 (1976), 273-284. | Zbl 0304.60004

[011] [P] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. | Zbl 0153.19101

[012] [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 | Zbl 0137.35301