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Springer Proceedings in Mathematics and Statistics Volume 33 For further volumes: http://www.springer.com/series/10533 Springer Proceedings in Mathematics and Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. Albert N. Shiryaev S. R. S. Varadhan Ernst L. Presman Editors Prokhorov and Contemporary Probability Theory In Honor of Yuri V. Prokhorov 123 Editors Albert N. Shiryaev Ernst L. Presman Steklov Mathematical Institute Central Economics and Math. Institute Russian Academy of Sciences Russian Academy of Sciences Moscow Moscow Russia Russia S. R. S. Varadhan Courant Institute New York University New York USA ISSN 2194-1009 ISSN 2194-1017 (electronic) ISBN 978-3-642-33548-8 ISBN 978-3-642-33549-5 (eBook) DOI 10.1007/978-3-642-33549-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012954482 Mathematical Subject Classification (2010): 60xx, 62xx, 60Fxx, 60Gxx, 81S25 c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This volume is dedicated to the prominent mathematician and leading expert on probability theory and mathematical statistics Yuri Vasilyevich Prokhorov, who celebrated his 80-th birthday on 15 December 2009. It consists of two parts. The first one contains papers written by his colleagues, friends and pupils who express their deep respect and sincerely love to him and his scientific activity. The second part contains two interviews with Yu.V. Prokhorov. The first inter- view was taken by Friedrich Gotze¨ and Willem R. van Zwet between November 13 and 28, 2006 at Bielefeld University. We decided to reproduce also the interview taken by Larry Shepp and published in Statistical Science 7 (1992), 123–130. Moscow, Russia Albert Shiryev New York, USA S.R.S. Varadhan Moscow, Russia Ernst Presman v Contents Yuri Vasilyevich Prokhorov ................................................... xi List of Publications by Yu.V. Prokhorov..................................... xix Part I Scientific Papers When Knowing Early Matters: Gossip, Percolation and Nash Equilibria ....................................................................... 3 David J. Aldous A Mathematical Model of Investment Incentives........................... 29 Vadim Arkin and Alexander Slastnikov The Shape of Asymptotic Dependence ....................................... 43 Guus Balkema, Paul Embrechts, and Natalia Nolde Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes ..................................... 69 Ole E. Barndorff-Nielsen, Jose´ Manuel Corcuera, and Mark Podolskij Retrieving Information from Subordination ................................ 97 Jean Bertoin and Marc Yor Asymptotic Expansions for Distributions of Sums of Independent Random Vectors ............................................. 107 Algimantas Bikelis An Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems .......................... 127 Alexander A. Borovkov and Konstantin A. Borovkov Optimal and Asymptotically Optimal Control for Some Inventory Models............................................................... 141 Ekaterina Bulinskaya vii viii Contents Levy Preservation and Associated Properties for f -Divergence Minimal Equivalent Martingale Measures .................................. 163 Suzanne Cawston and Lioudmila Vostrikova Non-standard Limit Theorems in Number Theory ........................ 197 Francesco Cellarosi and Yakov G. Sinai Additive Functions and Gaussian Measures ................................ 215 Linan Chen and Daniel W. Stroock Free Infinitely Divisible Approximations of n-Fold Free Convolutions .............................................................. 225 Gennadii Chistyakov and Friedrich Gotze¨ Accurate Approximation of Correlation Coefficients by Short Edgeworth-Chebyshev Expansion and Its Statistical Applications....... 239 Gerd Christoph, Vladimir V. Ulyanov, and Yasunori Fujikoshi The Stein-Tikhomirov Method and Berry-Esseen Inequality for Sampling Sums from a Finite Population of Independent Random Variables.............................................................. 261 Shakir K. Formanov and Tamara A. Formanova On One Inequality for Characteristic Functions ........................... 275 Nicko Gamkrelidze On the Nonlinear Filtering Equations for Superprocesses in Random Environment ......................................................... 281 Bronius Grigelionis Upper Bounds for Bernstein Basis Functions ............................... 293 Vijay Gupta and Tengiz Shervashidze On Distribution of Zeros of Random Polynomials in Complex Plane .... 303 Ildar Ibragimov and Dmitry Zaporozhets Dependence and Interaction in Branching Processes ...................... 325 Peter Jagers and Fima C. Klebaner Testing Functional Connection between Two Random Variables......... 335 Gyula O.H. Katona The Symmetry Group of Gaussian States in L2.Rn/ ....................... 349 Kalyanapuram R. Parthasarathy Solution of the Optimal Stopping Problem for One- Dimensional Diffusion Based on a Modification of the Payoff Function ......................................................................... 371 Ernst Presman Contents ix Part II Interviews The Times of Yuri Vasilyevich Prokhorov ................................... 407 Friedrich Gotze¨ and Willem R. van Zwet A Conversation with Yuri Vasilyevich Prokhorov .......................... 433 Larry Shepp Yuri Vasilyevich Prokhorov Yuri Vasil’evich Prokhorov was born in Moscow in the family of a highway construction engineer. In 1944 after finishing high school as an external student at the age of fourteen and a half, he entered the Bauman High School (now Moscow Technical University). In 1945 he voluntary transferred to the Faculty of Mechanics and Mathematics of Moscow State University and graduated in 1949. Right after graduating he became a research assistant at the Steklov Institute of Mathematics. In 1952 he got a PhD degree, and in 1956 defended a Doctorate dissertation in Physical and Mathematical Sciences. In 1956 and 1957 he taught at the Moscow Institute of Engineering Physics, and in 1958 he returned to the Steklov Institute, where he continued working since then, replaced A.N. Kolmogorov as the head of the Department of Probability Theory in 1961. In parallel with his work at the Steklov Institute Prokhorov has taught at Moscow State University. Initially he lectured at the Faculty of Mechanics and Mathematics, where he received the title of Professor in 1958. In 1970 he moved to the newly formed Faculty of Computational Mathematics and Cybernetics, where he continues to head the Mathematical Statistics Department. In 1966 Prokhorov was elected a Corresponding Member of Academy of Sciences of the USSR, and a Full Member in 1972. The scientific activity of Prokhorov began in his student years at the University. During his third year he started to actively participate in the probability theory seminar led by Academician Kolmogorov, and from that time he became one of Kolmogorov’s disciples for many years. In spring of 1948, being a fourth year student at the age of 18 Prokhorov wrote his first research paper which was published
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