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

H. Heyer and G. Pap

Journal of Theoretical Probability, 1997, vol. 10, issue 4, 1003-1052

Abstract: 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.

Keywords: Central limit theorem for Lie groups; noncommutative infinitesimal arrays of probability measures; convolution hemigroups; diffusion hemigroups; processes with independent increments (search for similar items in EconPapers)
Date: 1997
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1022670818425

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