Multiparameter Processes An Introduction to Random Fields
Multiparameter processes extend the existing one-parameter theory of random processes in an elegant way, and have found connections to diverse disciplines such as probability theory, real and functional analysis, group theory, analytic number theory, and
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Davar Khoshnevisan
Multiparameter Processes An Introduction to Random Fields
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Springer
Davar Khoshnevisan Department of Mathematics University of Utah Salt Lake City, Dt 84112-0090 [email protected]
With 12 illustrations. Mathematics Subject Classification (2000): 60Gxx, 60G60 Library of Congress Cataloging-in-Publication Data Khoshnevisan, Davar. Multiparameter processes: an introduction to random fields / Davar Khoshnevisan. p. cm. - (Springer monographs in mathematics) Includes bibliographical references and index ISBN 978-1-4419-3009-5 1. Random fields. I. Title. II. Series. QA274.45 .K58 2002 2002022927 519.2'3--dc21 Printed on acid-free paper. © 2002 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any fonn of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar tenns, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary lights. Manufacturing supervised by Jerome Basma. Camera-ready copy prepared from the author's LaTeX files.
9 8 7 654 3 2 I ISBN 978-1-4419-3009-5 ISBN 978-0-387-21631-7 (eBook) DOl 10.1007/978-0-387-21631-7 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH
Preface
This book aims to construct a general framework for the analysis of a large family of random fields, also known as multiparameter processes. The need for such a development was pointed out in Doob (1990, p. 47). Referring to the theory of one-parameter stochastic processes, Doob writes: 1 Our definition of a stochastic process is historically conditioned and has obvious defects. In the first place there is no mathematical reason for restricting T to be a set of real numbers, and in fact interesting work has already been done in other cases. (Of course, the interpretation of t as time must then be dropped.) In the second place there is no mathematical reason for restricting the value assumed by the Xt'S to be numbers. There are a number of compelling reasons for studying random fields, one of which is that, if and when possible, multiparameter processes are a natural extension of existing one-parameter processes. More exciting still are the various interactions between the theory of multiparameter processes and other disciplines, including probability itself. For example, in this book the reader will learn of various connections to real and functional analysis, a mo
