Get Probability Measures on Groups: Proceedings of the Sixth PDF

By Walter R. Bloom (auth.), Herbert Heyer (eds.)

ISBN-10: 3540115013

ISBN-13: 9783540115014

ISBN-10: 3540392068

ISBN-13: 9783540392064

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Read Online or Download Probability Measures on Groups: Proceedings of the Sixth Conference Held at Oberwolfach, Germany, June 28–July 4, 1981 PDF

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Additional info for Probability Measures on Groups: Proceedings of the Sixth Conference Held at Oberwolfach, Germany, June 28–July 4, 1981

Sample text

On the other hand, the m e t h o d s emerge m a i n l y from the theory of group representations. on limits of Especially, the theorems (the products of the r o w s ~f) t r i a n g u l a r arrays ~ m a l i s e d p o s i t i v e definite kernels on locally compact groups in ~3]) regains an i n t e r p r e t a t i o n as limit theorems dependent quantum mechanical random variables, for sums of in- a programmatic for w h i c h only a few hints can be given. Therefore, nor- (as claim, consider this report just as an invitation to look at the first elements of a theory of q u a n t u m m e c h a n i c a l limit theorems (which, by the way, has a more regular b e h a v i o u r as the classical counterpart, for instance, reflecting, the fact that the algebra of b o u n d e d operators has in many respects b e t t e r m a n n e r s c o m p a r e d w i t h an algebra of continuous functions).

4 ~ ( [B|f-gn,klPd~ )I/p + ~k 9 2~k and n similar ~ f l f - f k l d ~ 2~ k. Hence ~f-fk~p,1 < 4~ k holds for all k ~ , which shows f~LP'I(X,E). 5 it is not difficult to obtain the following characterization of the Banach spaces of Poisson type and cotype. 6 Theorem. (a) If (and only if) E is of Poisson type p ( 1 ~ p ~ ) then every Radon measure F on E+~{o] w i t h f N x U P F ( d x ) ~ and ~xF(dx)~E+ is a L~vy measure of an infinitely divisible measure of order p. (b) If (and only if) E is of Poisson cotype p ( 1 ~ p ~ ) then the L&vy measure F of every infinitely divisible measure of order p fulfills the conditions fllx~PF(dx)~ and )xF(dx) E+.

One can show with a more detailed analysis of the Laplace transforms that in the lattice case this shifting is normally unnecessary even if the sequence (Gn) is not increasing. 3 Proposition 9 Let (e(Gn)) be a sequence of Poisson measures on the cone E+ such that (e(Gn)@~ a ) converges for a certain sequence (an ) in E+ 9 Then (e(Gn)) converges n if and only if (u) lim sup sup s~o n~ x'~o,ux',~l ~ Wx,x'~Gn(dX ) = o . [lx~ If lim e(Gn)~e a = eo(F) then condition (u) is automatically satisfied. If E is w~akly sequentially complete then necessarily (e(Gn)) converges to eo(F ) relative to the weak topology on E.

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Probability Measures on Groups: Proceedings of the Sixth Conference Held at Oberwolfach, Germany, June 28–July 4, 1981 by Walter R. Bloom (auth.), Herbert Heyer (eds.)


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