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Dissipation of Convolution Powers in a Metric Group

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  • Wojciech Jaworski

    (Carleton University)

Abstract

In contrast to what is known about probability measures on locally compact groups, a metric group G can support a probability measure μ which is not carried on a compact subgroup but for which there exists a compact subset C⊆G such that the sequence μ n (C) fails to converge to zero as n tends to ∞. A noncompact metric group can also support a probability measure μ such that supp μ=G and the concentration functions of μ do not converge to zero. We derive a number of conditions which guarantee that the concentration functions in a metric group G converge to zero, and obtain a sufficient and necessary condition in order that a probability measure μ on G satisfy lim n→∞ μ n (C)=0 for every compact subset C⊆G.

Suggested Citation

  • Wojciech Jaworski, 2007. "Dissipation of Convolution Powers in a Metric Group," Journal of Theoretical Probability, Springer, vol. 20(3), pages 487-503, September.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:3:d:10.1007_s10959-007-0072-3
    DOI: 10.1007/s10959-007-0072-3
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    References listed on IDEAS

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    1. Wojciech Jaworski, 1997. "Contractive Automorphisms of Locally Compact Groups and the Concentration Function Problem," Journal of Theoretical Probability, Springer, vol. 10(4), pages 967-989, October.
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