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Limit theorems on locally compact Abelian groups. (English) Zbl 1152.60011

Summary: We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of J. Gaiser’s theorem [Konvergenz stochastischer Prozesse mit Werten in einer lokalkompakten Abelschen Gruppe, Dissertation, Universität Tübingen (1994; Zbl 0834.60013), Satz 1.3.6], since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure cannot have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of \(p\)-adic integers and the \(p\)-adic solenoid.

MSC:

60B10 Convergence of probability measures
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization

Citations:

Zbl 0834.60013
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References:

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