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Speed of Convergence of Transformed Convolution Powers of a Probability Measure on a Compact Connected Group

In: Probability Measures on Groups X

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  • Peter Eisele

    (Katholische Universität Eichstätt)

Abstract

In [1] Bhattacharya has shown that the sequence of the n-fold convolution powers of a probability measure μ on a compact connected group G converges in the norm exponentially fast to the normed Haar measure ω G if μ k nonzero ω G -absolutely continuous part for some k ∈ N. Our main result is that this holds true also for convolution products ϱ n = λ * T(λ) * • • • * T n-1(λ) where A is a probability measure on G and T a continuous automorphism of G. As an auxiliary result we prove that for an arbitrary nonempty open subset B of G the sequence B n := BT(B) • • • T n-1 (B) reaches G after finitely many steps. The results concerning ϱ n are applied to sequences μ n x -n of shifted convolution powers on an arbitrary topological group.

Suggested Citation

  • Peter Eisele, 1991. "Speed of Convergence of Transformed Convolution Powers of a Probability Measure on a Compact Connected Group," Springer Books, in: Herbert Heyer (ed.), Probability Measures on Groups X, pages 117-127, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4899-2364-6_8
    DOI: 10.1007/978-1-4899-2364-6_8
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