Convergence of noncommutative triangular arrays of probability measures on a Lie group

Heyer, Herbert and Pap, Gyula (1997) Convergence of noncommutative triangular arrays of probability measures on a Lie group. Journal of Theoretical Probability, 10 (4). pp. 1003-1052. ISSN 0894-9840

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Abstract

A measure-theoretic approach to the central limit problem for noncommutative infinitesimal arrays of random variables taking values in a Lie group G is given. Starting with an array {mu(nl): (n, l) is an element of N-2} of probability measures on G and instance 0 less than or equal to x less than or equal to t one forms the finite convolution products mu(n)(s, t): = mu(n,kn(s)+1)*...*mu(n,kn(t)). The authors establish sufficient conditions in terms of Levy-Hunt characteristics for the sequence {mu(n)(s, t): n is an element of N} to converge towards a convolution hemigroup (generalized semigroup) of measures on G which turns out to be of bounded variation. In particular, conditions are stated that force the limiting hemigroup to be a diffusion hemigroup. The method applied in the proofs is based on properly chosen spaces of differentiable Functions and on the solution of weak backward evolution equations on G.

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