# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Local conditional expectations and an application to a central limit theorem on a locally compact Abelian group. (English) Zbl 0633.60013

Let G be a locally compact second countable Abelian group with dual group $\stackrel{^}{G}$ and let g be a local inner product on $G×\stackrel{^}{G}$ in the sense of 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 $\left({\Omega },ℱ,P\right)$ there exists a local conditional expectation $\overline{X}$ of X given a sub- $\sigma$-field $𝒜$ of $ℱ$; i.e. $\overline{X}$ is a G-valued $𝒜$-measurable random variable on $\left({\Omega },ℱ,P\right)$ such that $<\overline{X},y>=exp\left\{i𝔼\left(g\left(X,y\right)|𝒜\right\}$ for all $y\in \stackrel{^}{G}$ (Theorem 1).

Thereafter this concept is applied to the convergence of adapted triangular G-valued arrays $\left\{{S}_{nj},{ℱ}_{nj}:$ $1\le j\le {k}_{n}$, $n\ge 1\right\}$. Under appropriate assumptions on the differences ${X}_{nj}={S}_{nj}-{S}_{nj-1}$ and their local conditional expectations ${\overline{X}}_{nj}$ given ${ℱ}_{n,j-1}$, the sequence ${\left({S}_{n{k}_{n}}\right)}_{n\ge 1}$ 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)].

Reviewer: E.Siebert

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60F15 Strong limit theorems 60F05 Central limit and other weak theorems
##### References:
 [1] Bingham, M.S.: A central limit theorem for approximate martingale arrays with values in a locally compact abelian group. Math. Z.192, 409-419 (1986) · Zbl 0599.60010 · doi:10.1007/BF01164015 [2] Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. New York: Academic Press 1980 [3] Heyer, H.: Probability Measures on Locally Compact Groups. Berlin Heidelberg New York: Springer 1977 [4] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, Volumes I and II. Berlin Heidelberg New York: Springer 1963, 1970 [5] Neveu, J.: Mathematical Foundations of the Calculus of Probability. San Francisco: Holden-Day 1965 [6] Parthasarathy, K.R.: Probability Measures on Metric Spaces. New York: Academic Press 1967 [7] Rényi, A.: On stable sequences of events. Sanky? Ser. A25, 293-302 (1963) [8] Rudin, W.: Fourier analysis on groups. New York: Wiley (interscience) 1962