Local conditional expectations and an application to a central limit theorem on a locally compact Abelian group (Q1096233)

Let G be a locally compact second countable Abelian group with dual group \(\hat G\) and let g be a local inner product on \(G\times \hat G\) in the sense of \textit{K. R. Parthasarathy} [see: Probability measures on metric spaces (1967; Zbl 0153.191)]. At first it is shown that for any G-valued random variables X on a probability space \((\Omega,{\mathcal F},P)\) there exists a local conditional expectation \(\bar X\) of X given a sub- \(\sigma\)-field \({\mathcal A}\) of \({\mathcal F}\); i.e. \(\bar X\) is a G-valued \({\mathcal A}\)-measurable random variable on \((\Omega,{\mathcal F},P)\) such that \(<\bar X,y>=\exp \{i{\mathbb{E}}(g(X,y)| {\mathcal A}\}\) for all \(y\in \hat G\) (Theorem 1). Thereafter this concept is applied to the convergence of adapted triangular G-valued arrays \(\{S_{nj},{\mathcal F}_{nj}:\) \(1\leq j\leq k_ n\), \(n\geq 1\}\). Under appropriate assumptions on the differences \(X_{nj}=S_{nj}-S_{nj-1}\) and their local conditional expectations \(\bar X_{nj}\) given \({\mathcal F}_{n,j-1}\), the sequence \((S_{nk_ n})_{n\geq 1}\) converges stably in law to a mixture of Gaussian distributions on G (Theorem 2). The proof of this central limit theorem applies a previous version of a theorem due to the same author [Math. Z. 192, 409-419 (1986; Zbl 0599.60010)].