Let G be a locally compact second countable Abelian group with dual group and let g be a local inner product on 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 there exists a local conditional expectation of X given a sub- -field of ; i.e. is a G-valued -measurable random variable on such that for all (Theorem 1).
Thereafter this concept is applied to the convergence of adapted triangular G-valued arrays , . Under appropriate assumptions on the differences and their local conditional expectations given , the sequence 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)].