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Convergence of Noncommutative Triangular Arrays of Probability Measures on a Lie Group

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  • H. Heyer
  • G. Pap

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 $$\{ \mu _{n\ell } :(n,\ell ) \in \mathbb{N}^2 \} $$ of probability measures on G and instance 0≤s≤t one forms the finite convolution products $$\mu _n (s,t): = \mu _{n,k_n (s) + 1} * \cdots *\mu _{n,k_n (t)} $$ . The authors establish sufficient conditions in terms of Lévy-Hunt characteristics for the sequence $$\{ \mu _n (s,t):n \in \mathbb{N}\} $$ 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 difierentiable functions and on the solution of weak backward evolution equations on G.

Suggested Citation

  • H. Heyer & G. Pap, 1997. "Convergence of Noncommutative Triangular Arrays of Probability Measures on a Lie Group," Journal of Theoretical Probability, Springer, vol. 10(4), pages 1003-1052, October.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:4:d:10.1023_a:1022670818425
    DOI: 10.1023/A:1022670818425
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    Cited by:

    1. Peter Becker-Kern, 2003. "Stable and Semistable Hemigroups: Domains of Attraction and Self-Decomposability," Journal of Theoretical Probability, Springer, vol. 16(3), pages 573-598, July.

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