Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory
- PDF / 5,760,242 Bytes
- 114 Pages / 504 x 720 pts Page_size
- 83 Downloads / 189 Views
272 K. R. Parthasarathy University of Bombay, Bombay/India
K. Schmidt Bedford College, University of London, London/England
Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory
Springer-Verlag Berlin . Heidelberg . New York 1972
AMS Subject Classifications (1970): 43A05, 43A35, 43A65, 46ClO, 6OBI5, 6OF05
ISBN 3-540-05908-3 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05908-3 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. !.Imler ),1 01 tilt' German C(,pyrighr LJW where, 0p,e, ure madt I;'r other ilun :1 fee j, ble tH til(' the amount of the fee to be determined by agreement with the publisher.
©> by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card-Number 72-85400. Printed in Germany.
Offsetdruek: Julius Beltz, HemsbachlBergstr.
PREFACE
These notes are mainly based on a course of lectures given by the first named author at the Research and Training School of the Indian Statistical Institute, Calcutta during May 1971. A first and slightly shorter version of these notes has appeared under the same title as publication No. M71-1 of the Research and Training School of the Indian Statistical Institute. Some of the results were obtained when the authors were at the Statistical Laboratory, Mathematics Department, University of Manchester, in 1970. The notion of a continuous tensor product of Hilbert spaces and group representations appears in the work of H. H. Araki and E.J. Woods [1] and R.F. Their analysis leads to a connection between continuous tensor products and the theory of infinitely divisible distributions of Probability theory. The present work contains a systematic study of these notions in terms of positive definite kernels with invariance properties under a group action. Such analysis also leads to a unified approach to the central limit problems of Probability theory, the theory of stochastic processes with stationary increments and construction of free fields in Quantum Mechanics. The contents of these notes are divided into three parts. In part 1 the notion of a projectively invariant positive definite kernel on an abstract G-space is introduced and obtained as expectation value of a projective unitary representation of the group G. Affine invariant conditionally positive definite kernels are investigated and a representation of such kernels in terms of unitary representations and first order cocycles is obtained. Using these ideas and the theory of multiplicative measures, continuous tensor products of Hilbert spaces and representations are constructed. The Fock-Cook construction of the Bose-Einstein field in Quantum Mechanics [4] , arises as a natural consequence of this theory. Much of the inspiration for this approach was derived
Data Loading...
