General solution of the functional central limit problems on a Lie group

Pap, Gyula (2004) General solution of the functional central limit problems on a Lie group. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 7 (1). pp. 43-87. ISSN 0219-0257

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Abstract

Starting with an array (mu(n),e)n,eis an element ofN of probability measures on a Lie group G, one forms the convolution products mun (s, t) := mu(n),k(n) (s)+l * ...*mu(n),k(n)(t), 0 less than or equal to s less than or equal to t, where kn : R+ --> Z(+) are increasing scaling functions. Sufficient conditions are established for convergence mu(n)(s, t) --> mu(s, t) as n --> infinity, where (mu(s, t))(O) less than or equal to s less than or equal to t is necessariby a convolution hemigroup, i.e. mu(s, t) = mu(s, r) * mu(r, t), 0 less than or equal to s less than or equal to r less than or equal to t.Using previous results concerning convolution hemigroups of finite variation (Heyer and Pap(9)), this leads to a bijection between the set of convolution hemigroups and an appropriate parameter set containing time-dependent Levy-Khinchin type triplets. Hence, at the same time, we parametrize the set of measures generated by stochastically continuous processes with independent left increments in G. The above-mentioned sufficient conditions turn out to be necessary and sufficient for convergence of a sequence of random step functions corresponding to the triangular array towards a stochastically continuous process with independent left increments. Connections with results of Feinsilver(4) and Kunita(11) are discussed.

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