GENERAL SOLUTION OF THE FUNCTIONAL CENTRAL LIMIT PROBLEMS ON A LIE GROUP

Abstract

Starting with an array (μn,ℓ)n,ℓ∈ℕ of probability measures on a Lie group G, one forms the convolution products μn(s,t):=μn,kn(s)+1*⋯*μn,kn(t), 0≤s≤t, where kn:ℝ+→ℤ+ are increasing scaling functions. Sufficient conditions are established for convergence μn(s,t)→μ(s,t) as n→∞, where (μ(s,t))0≤s≤t is necessariliy a convolution hemigroup, i.e. μ(s,t)=μ(s,r)*μ(r,t), 0≤s≤r≤t. Using previous results concerning convolution hemigroups of finite variation (Heyer and Pap9), this leads to a bijection between the set of convolution hemigroups and an appropriate parameter set containing time-dependent Lévy–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 Feinsilver4 and Kunita11 are discussed.