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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 G ^ and let g be a local inner product on G×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 (Ω,,P) there exists a local conditional expectation X ¯ of X given a sub- σ-field 𝒜 of ; i.e. X ¯ is a G-valued 𝒜-measurable random variable on (Ω,,P) such that <X ¯,y>=exp{i𝔼(g(X,y)|𝒜} for all yG ^ (Theorem 1).

Thereafter this concept is applied to the convergence of adapted triangular G-valued arrays {S nj , nj : 1jk n , n1}. Under appropriate assumptions on the differences X nj =S nj -S nj-1 and their local conditional expectations X ¯ nj given n,j-1 , the sequence (S nk n ) n1 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:
60B15Probability measures on groups or semigroups, Fourier transforms, factorization
60F15Strong limit theorems
60F05Central 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