By Bingham M.S.

A crucial restrict theorem is given for uniformly infinitesimal triangular arrays of random variables within which the random variables in each one row are exchangeable and take values in a in the community compact moment countable abclian staff. The proscribing distribution within the theorem is Gaussian.

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**Example text**

9) a(z;glg2)= a(z;gl)u(xgl;gz);for all gl;g2 E G , x E X . 10) does give a representation of G. Furthermore, if u is continuousfunitary and X has a Haar measure d x , then T , also becomes a continuousfunitary representation'' of G on space L 2 ( X ; Y ) , of square-integrable Y-valued functions on X (problem 9). 2. Regular representations R on G and quotients X = H\G, are induced by the trivial cocycle u, R = ind(1 I H;G). g. continuous, differentiable, L2-functions, etc. 7) to make T unitary.

In particular, unitary (or more general bounded) representations, I(T,(I5 C, yield bounded operators {T,} for all f in the group algebra with weight w, L = L'(G), indeed, I)T,(II Cllflll. 2. Regular and induced representation 33 R f [ h ]= f*h. 15) has to do with nice (smoothing/regularizing) properties of operators { T I } . To wit the “unitary/bounded group operators” {T,} could yield a better class of “algebra operators” { Tf}:compact, Hilbert-Schmidt, or trace-class. The latter often depends on the regularity properties of L = em(G);P‘(G), the “smoother11 { f ’ s } one takes the “better” { T I }results.

Regular and induced representation 33 R f [ h ]= f*h. 15) has to do with nice (smoothing/regularizing) properties of operators { T I } . To wit the “unitary/bounded group operators” {T,} could yield a better class of “algebra operators” { Tf}:compact, Hilbert-Schmidt, or trace-class. The latter often depends on the regularity properties of L = em(G);P‘(G), the “smoother11 { f ’ s } one takes the “better” { T I }results. A simple illustration is furnished by convolutions with f E Cm(G) (or Coo).