𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Limit theorems on locally compact Abelian groups

✍ Scribed by Mátyás Barczy; Alexander Bendikov; Gyula Pap

Publisher
John Wiley and Sons
Year
2008
Tongue
English
Weight
221 KB
Volume
281
Category
Article
ISSN
0025-584X

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

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 Gaiser's theorem [4, 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

📜 SIMILAR VOLUMES

Fourier type 2 operators with respect to
Fourier type 2 operators with respect to locally compact abelian groups
✍ Aicke Hinrichs; Mariusz Piotrowski 📂 Article 📅 2004 🏛 John Wiley and Sons 🌐 English ⚖ 121 KB

## Abstract A linear and bounded operator __T__ between Banach spaces __X__ and __Y__ has Fourier type 2 with respect to a locally compact abelian group __G__ if there exists a constant __c__ > 0 such that∥__T__$\hat f$∥~2~ ≤ __c__∥__f__∥~2~ holds for all __X__‐valued functions __f__ ∈ __L__^__X__^

Littlewood–Paley and Multiplier Theorems
Littlewood–Paley and Multiplier Theorems on Weighted Spaces over Locally Compact Vilenkin Groups
✍ T.S. Quek 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 222 KB

We extend the Littlewood᎐Paley theorem to L G , where G is a locally w compact Vilenkin group and w are weights satisfying the Muckenhoupt A p condition. As an application we obtain a mixed-norm type multiplier result on p Ž . L G and prove the sharpness of our result. We also obtain a sufficient co

Generalised sk-Spline Interpolation on C
Generalised sk-Spline Interpolation on Compact Abelian Groups
✍ J. Levesley; A.K. Kushpel 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 178 KB

The notion of sk-spline is generalised to arbitrary compact Abelian groups. A class of conditionally positive definite kernels on the group is identified, and a subclass corresponding to the generalised sk-spline is used for constructing interpolants, on scattered data, to continuous functions on th

Semisimple Banach Algebras Generated by
Semisimple Banach Algebras Generated by Strongly Continuous Representations of Locally Compact Abelian Groups
✍ G. Muraz; V.Q. Phong 📂 Article 📅 1994 🏛 Elsevier Science 🌐 English ⚖ 207 KB

We prove that if \(T\) is a strongly based continuous bounded representation of a locally compact abelian group \(G\) on a Banach Space \(X\), and if the spectrum of \(T\) is countable, then the Banach algebra generated by \(f(T)=\int_{G} f(g) T(g) d g\), \(f \in L^{1}(G)\), is semisimple. 1994 Acad