A central limit theorem for approximate martingale arrays with values in a locally compact Abelian group
Let G be a locally compact second countable Abelian group with dual \(\hat G\) and let g be a local inner product on \(G\times \hat G\). Suppose that \(\{S_{nj},{\mathcal F}_{nj}:1\leq j\leq k_ n,n\geq 1\}\) is an adapted triangular array of G-valued random variables on a probability space (\(\Omega\),\({\mathcal F},P)\) with differences \(\{X_{nj}\}\); thus \(\{{\mathcal F}_{nj}:1\leq j\leq k_ n\}\) is a filtration in \({\mathcal F}\) for each n, \(S_{nj}\) is \({\mathcal F}_{nj}\)-measurable and \(S_{nj}=\sum^{j}_{k=1}X_{nk}\). Let \(\Phi: \hat G\times \Omega \to {\mathbb{R}}\) be a random continuous nonnegative quadratic form. Theorem: Assume that, for every neighbourhood N of the identity in G, \[ P(X_{nj}\not\in N \text{ for some \(j=1,2,...,k_ n)\to 0\) as }n\to \infty \] and that, for every \(y\in \hat G\), \[ \sum^{k_ n}_{j=1}| E[g(X_{nj},y)| {\mathcal F}_{n,j-1}]| \to 0\text{ in probability} \] and \[ \sum^{k_ n}_{j=1}g(X_{nj},y)^ 2\to \Phi (y)\text{ in probability as } n\to \infty. \] If in addition the filtrations are nested (i.e. \({\mathcal F}_{n,j}\subseteq {\mathcal F}_{n+1,j}\) for all n,j) then, for every \(F\in {\mathcal F}\) with \(P(F)>0\), the conditional distribution of \(S_{nk_ n}\) given F converges weakly as \(n\to \infty\) to the mixture of Gaussian distributions with characteristic function \(y\to E(\exp [-\Phi (y)]| F)\), \(y\in \hat G\).
scientific article; zbMATH DE number 3967545