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Zbl 0633.60013
Bingham, Michael S.
Local conditional expectations and an application to a central limit theorem on a locally compact Abelian group.
(English)
[J] Math. Z. 195, 1-12 (1987). ISSN 0025-5874; ISSN 1432-1823/e

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 {\it 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 $(\Omega,{\cal F},P)$ there exists a local conditional expectation $\bar X$ of X given a sub- $\sigma$-field ${\cal A}$ of ${\cal F}$; i.e. $\bar X$ is a G-valued ${\cal A}$-measurable random variable on $(\Omega,{\cal F},P)$ such that $<\bar X,y>=\exp \{i{\bbfE}(g(X,y)\vert {\cal A}\}$ for all $y\in \hat G$ (Theorem 1). \par Thereafter this concept is applied to the convergence of adapted triangular G-valued arrays $\{S\sb{nj},{\cal F}\sb{nj}:$ $1\le j\le k\sb n$, $n\ge 1\}$. Under appropriate assumptions on the differences $X\sb{nj}=S\sb{nj}-S\sb{nj-1}$ and their local conditional expectations $\bar X\sb{nj}$ given ${\cal F}\sb{n,j-1}$, the sequence $(S\sb{nk\sb n})\sb{n\ge 1}$ 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)].
[E.Siebert]
MSC 2000:
*60B15 Probability measures on groups
60F15 Strong limit theorems
60F05 Weak limit theorems

Keywords: triangular array; mixture of Gaussian distributions; locally compact second countable Abelian group; local conditional expectation; central limit theorem

Citations: Zbl 0153.191; Zbl 0599.60010

Cited in: Zbl 0639.60007

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