ISSN:
1572-9230
Keywords:
Central limit theorem for Lie groups
;
noncommutative infinitesimal arrays of probability measures
;
convolution hemigroups
;
diffusion hemigroups
;
processes with independent increments
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022670818425
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