“Four legs good, two legs better ” A modified version of the Animal Farm’s Constitution “Two logs good, p logs better ” The original Constitution of mathematicians RAN D O M WALK IN RANDOM AND N O N- RAND O M ENVIRONMENTS h E C D N D E D I T I O N This page intentionally left blank L RANDEOM WALK IN RANDOM AND N NON RANDOM N M ENVIRONMENTS ENVIRO E C O N EDITION D Pal Revesz Technische Universitat Wien, Austria Technical University of Budapest, Hungary N E W JERSEY * LONDON * World Scientific ;- SINGAPORE BEIJING * SHANGHAI HONG KONG * TAIPEI - CHENNAI Published by World Scientific Publishing Co Pte.Ltd Toh Tuck Link, Singapore 596224 USA ofice; 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Random walk in random and non-random environments / Pfll RCvCsz. 2nd ed p cm Includes bibliographical references and indexes ISBN 981-256-361-X (alk paper) Random walks (Mathematics) I Title QA274.73 R48 2005 19.2’82 dc22 2005045536 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright 2005 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher Printed in Singapore by Mainland Press Preface to the First Edition “I did not know that it was so dangerous to drink a beer with you You write a book with those you drink a beer with,” said Professor Willem Van Zwet, referring to the preface of the book Csorgo and I wrote (1981) where it was told that the idea of that book was born in an inn in London over a beer In spite of this danger Willem was brave enough t o invite me t o Leiden in 1984 for a semester and to drink quite a few beers with me there In fact I gave a seminar in Leiden, and the handout of that seminar can be considered as the very first version of this book I am indebted to Willem and to the Department of Leiden for a very pleasant time and a number of useful discussions I wrote this book in 1987-89 in Vienna (Technical University) partly supported by Fonds zur Forderung der Wissenschaftlichen Forschung, Project Nr P6076 During these years I had very strong contact with the Mathematical Institute of Budapest I am especially indebted t o Professors E Csaki and A Foldes for long conversations which have a great influence on the subject of this book The reader will meet quite often with the name of P Erdos, but his role in this book is even greater Especially most results of Part I1 are fully or partly due to him, but he had a significant influence even on those results that appeared under my name only Last but not least, I have t o mention the name of M Csorgo, with whom I wrote about 30 joint papers in the last 15 years, some of them strongly connected with the subject of this book P Rkvksz Technical University of Vienna Wiedner Hauptstrasse 8-10/107 -4-1040 Vienna Austria Vienna, 1989 V This page intentionally left blank Preface to the Second Edition If you write a monograph on a new, just developing subject, then in the next few years quite a number of brand-new papers are going t o appear in your subject and your book is going t o be outdated If you write a monograph on a very well-developed subject in which nothing new happens, then it is going t o be outdated already when it is going to appear In 1989 when I prepared the First Edition of this book it was not clear for me that its subject was already overdeveloped or it was a still developing area A year later Erd6s told me that he had been surprised to see how many interesting, unsolved problems had appeared in the last few years about the very classical problem of coin-tossing (random walk on the line) In fact Erdos himself proposed and solved a number of such problems I was happy to see the huge number of new papers (even books) that have appeared in the last 16 years in this subject I tried t o collect the most interesting ones and to fit them in this Second Edition Many of my friends helped me to find the most important new results and to discover some of the mistakes in the First Edition My special thanks t o E CsAki, M Csorgo”,A Foldes, D Khoshnevisan, Y Peres, Q M Shao, B T6th, Z Shi Vienna, 2005 vii This page intentionally left blank Contents Preface to the First Edition V Preface to the Second Edition vii Introduction xv I SIMPLE SYMMETRIC RANDOM WALK IN Z’ Notations and abbreviations Introduction of Part I 1.1 Randomwalk 1.2 Dyadic expansion 1.3 Rademacher functions 1.4 Coin tossing 1.5 The language of the probabilist 9 10 10 11 11 Distributions 2.1 Exact distributions 2.2 Limit distributions 13 13 19 Recurrence and the Zero-One Law 3.1 Recurrence 3.2 The zero-one law 23 23 25 F’rom the Strong Law of Large Numbers to the Law of Iterated Logarithm 27 4.1 Borel-Cantelli lemma and Markov inequality 27 4.2 The strong law of large numbers 28 4.3 Between the strong law of large numbers and the law of iterated logarithm 29 4.4 The LIL of Khinchine 31 Lbvy Classes 5.1 Definitions 5.2 EFKPLIL 5.3 The laws of Chung and Hirsch 5.4 When will S, be very large? ix 33 33 34 39 39 366 REFERENCES HANSON, D L - RUSSO, R P (1983/A) Some results on increments of the Wiener process with applications t o lag sums of I.I.D random variables The Annals of Probability 11,609-623 (1983/B) Some more results on increments of the Wiener process The Annals of Probability 11, 1009-1015 HARTMAN, P - WINTNER, A (1941) On the law of iterated logarithm Amer J Math 63,169-176 HAUSDORFF, F (1913) Grundziige der Mengenlehre Leipzig HIRSCH, W M (1965) A strong law for the maximum cumulative sum of independent random variables Comm Pure Appl Math 18, 109-217 HOUGH, J B - PERES, Y (2005) An LIL for cover times of discs by planar random walk and Wiener sousage To appear HU, Y - SHI, Z (1998/A) The local time of simple random walk in random environment J Theoretical Probability 11, 765-793 (1998/B) The limits of Sinai’s simple random walk in random environment The Annals of Probability 26,1477-1521 IMHOF, I P (1984) Density factorizations for Brownian motion meander and the three-dimensional Bessel process J Appl Probab 21, 500-510 ITO, K (1942) Differential equations determining a Markoff process Kiyosi It6 Selected Papers Springer-Verlag, New York (1986), 42-75 ITO, K MCKEAN Jr., H P (1965) Diffusion processes and their sample paths Die Grundlagen der Mathematischen Wissenschaften Band 125 Springer-Verlag, Berlin - JAIN, N C - PRUITT, W E (1971) The range of transient random walk J Analyse Math 24, 369-373 (1972/A) The law of iterated logarithm for the range of random walk Ann Math Statist 43,1692-1697 REFERENCES 367 (1972/B) The range of random walk Proc Sixth Berkeley Symp Math Statist Probab 3,Univ California Press, Berkeley, 31-50 (1974) Further limit theorems for the range of random walk J Analyse Math 27,94-117 KALIKOW, S A (1981) Generalized random walk in a random environment The Annals of Probability 9, 753-768 KARLIN, S - OST, F (1988) Maximal length of common words among random letter sequences The Annals of Probability 16,535-563 KESTEN, H (1965) An iterated logarithm law for the local time Duke Math J 32, 447-456 (1980) The critical probability of band percolation on iZ2 equals 1/2 Comm Math Phys 74,41-59 (1986) The limit distribution of Sinai's random walk in random environment Comm Math Phys 138,299-309 (1987) Hitting probabilities of random walks on Z d Stochastic Processes and their Applications 25, 165-184 (1988) Recent progress in rigorous percolation theory Aste'risque 157158,217-231 KESTEN, H - SPITZER, F (1979) A limit theorem related to a new class of self similar processes Wahrscheinlichkeitstheorie uerw Gebiete 50,5-25 KEY, E S (1984) Recurrence and transience criteria for random walk in a random environment The Annals of Probability 12,529-560 KHINCHINE, A (1923) Uber dyadische Briiche Math Zeitschrift 18, 109-116 KHOSHNEVISAN, D (1994) Exact rates of convergence to Brownian local time The Annals of Probability 22,1295-1330 (2002) Multiparameter Processes Springer-Verlag, New York, Berlin, Heidelberg KNIGHT, F B (1981) Essentials of Brownian Motion and Diffusion Am Math SOC., Providence, R.I 368 REFERENCES (1986) On the duration of the longest excursion Seminar on Stochastic Processes, 1985 Birkhauser, Boston 117-147 KOLMOGOROV, A N (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung Springer, Berlin KOMLOS, J - MAJOR, P TUSNADY, G (1975) An approximation of partial sums of independent R.V.’s and the sample DF I Wahrscheinlichkeitstheorie verw Gebiete 32, 111-131 ~ (1976) An approximation of partial sums of independent R.V.3 and the sample DF 11 Wahrscheinlichkeitstheorie verw Gebiete 34, 33-58 LACEY, M T - PHILIPP, W (1990) A note on the almost sure central limit theorem Statistics Probability Letters 9, 201-205 d LAMPERTI, J (1977) Stochastic Processes A Survey of the Mathematical Theory Springer - Verlag, New York LAWLER, G F (1980) A self-avoiding random walk Duke Mathematical Journal 47, 655-692 (1991) Intersections of Random Walks Birkhauser, Boston (1993) On the covering time of a disc by simple random walk in two-dimensionals Seminar on Stochastic Processes 1992 Birkhauser, Boston 33, 189-208 LE GALL, J.-F (1986) PropriCtCs d’ intersections des marches alCatoires Convergence rers le temps local d’ intersection Comm Math Phys 104,471-507 (1988) Fluctuation results for the Wiener sausage The Annals of Probability 16, 991-1018 LE GALL, J.-F - ROSEN, J (1991) The range of stable random walks The Annals of Probability 19, 650-705 LEVY, P (1948) Processu Stochastique et Mouvement Brownien Gauthier lars, Paris - Vil- MAJOR, P (1988) On the set visited once by a random walk Probab Th Rel Fields 77,117-128 REFERENCES 369 MARCUS, M B - ROSEN, J (1997) Laws of iterated logarithm for intersections of random walks on Z4 Ann Inst H Poincare‘ Probab Statist 33, 37-63 MCKEAN Jr, H P (1969) Stochastic Integrals Academic Press, New York MOGUL’SKII, A A (1979) On the law of the iterated logarithm in Chung’s form for functional spaces Th of Probability and its Applications 24, 405-412 MORI, T (1989) More on the waiting time till each of some given patterns occurs as a run Preprint MUELLER, C (1983) Strassen’s law for local time Wahrscheinlichkeitstheorie verw Gebiete 63, 29-41 NEMETZ, T - KUSOLITSCH, N (1982) On the longest run of coincidences Wahrscheinlichkeitstheorie verw Gebiete 61, 59-73 NEWMAN, D (1984) In a random walk the number of “unique experiences” is two on the average S I A M Review 26, 573-574 OREY, S - PRUITT, W E (1973) Sample functions of the N-parameter Wiener process The A n nals of Probability 1, 138-163 ORTEGA, I - WSCHEBOR, M (1984) On the increments of the Wiener process Wahrscheinlichkeitstheorie verw Gebiete 65, 329-339 PERKINS, E (1981/A) A global instrinsic characterization of Brownian local time The Annals of Probability , 800-817 (1981/B) On the iterated logarithm law for local time Proc Amer Math SOC.81, 470-472 PERKINS, E - TAYLOR, S J (1987) Uniform measure results for the image of subsets under Brownian motion Probab Th Rel Fields 76, 257-289 370 REFERENCES PETROV, V V (1965) On the probabilities of large deviations for sums of independent random variables Th of Probability and its Applications 10,287-298 PETROWSKY, I G (1935) Zur ersten Randwertaufgabe der Warmleitungsgleichung Comp Math B 1, 383-419 PITT, J H (1974) Multiple points of transient random walk Proc Amer Math SOC.43, 195-199 POLYA, G (1921) Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz Math Ann 84,149-160 QUALLS, G - WATANABE, H (1972) Asymptotic properties of Gaussian processes Statistics 43,580-596 Annals Math RENYI, A (1970/A) Foundations of Probability Holden-Day, San Francisco (1970/B) Probability Theory AkadCmiai Kiad6, Budapest and NorthHolland, Amsterdam REVESZ, P (1978) Strong theorems on coin tossing PTOC Int Cong of Mathematicians, Helsinki (1979) A generalization of Strassen's functional law of iterated logarithm Wahrscheinlichkeitstheorie verw Gebiete 50,257-264 (1981) Local time and invariance Lecture Notes in Math.: Analytical Methods i n Probab Th 861,128-145 (1982) On the increments of Wiener and related process The Annals of Probability 10,613-622 (1988) In random environment the local time can be very big SociCtL Mathkmatique de France, Aste'risque 157-158, 321-339 (1989) Simple symmetric random walk in Zd Almost Everywhere Convergence Proceedings of the Int Conf on Almost Everywhere Convergence (ed G A Edgar, L Sucheston) Academic Press, Boston 369-392 (1990/A) Estimates of the largest disc covered by a random walk The Annals of Probability 18,1784-1789 (1990/B) On the volume of spheres covered by a random walk A tribute to Paul Erd6s (ed A Baker, B Bollobb, A Hajnal) Cambridge Univ Press 341-347 REFERENCES 371 (1991) Waiting for the coverage of the Strassen's set Studia Sci Math Hung 26, 379-391 (1992) Black holes on the plane drawn by a Wiener process Probability Theory and Related Fields 93,21-37 (1993/A) Clusters of a random walk on the plane The Annals of Probability 21, 318-328 (1993/B) Covering problems Theory of Probability and its Applications 38,367-379 (1993/C) A homogenity property of the random walk Acta Sci ' Math (Szeged) 57,477-484 (1996) Balls left empty by a critical branching Wiener process J of Applied Math and Stochastic Analysis 9,531-549 (2000) On the inverse local time process of a plane random walk Periodica Math Hung 41,227-236 (2004) The maximum of the local time of a transient random walk Studia Sci Math Hung 41, 379-390 RGVESZ, P - SHI, Z (2000) Strong approximation of spatial random walk in random scenery Stochastic Processes and their Applications 88,329-345 RIESZ, F - SZ NAGY, B (1953) F'unctional Analysis Frederick Ungar, New York ROSEN, J (1997) Laws of the iterated logarithm for triple intersections of threedimensional random walks Electron J Probab 2, 1-32 SAMAROVA, S S (1981) On the length of the longest head-run for a Markov chain with two states Th of Probability and its Applications 26, 489-509 SCHATTE, P (1988) On strong versions of the central limit theorem Math Nachr 137,249-256 SHAO, Q M (1995) On a conjecture of R6vCsz Proc A m Math SOC.123,575-582 SHI, Z - TOTH, B (2000) Favourite sites of simple random walks Periodica Math Hung 41, 237-249 372 REFERENCES SIMONS, G (1983) A discrete analogue and elementary derivation of “LCvy’s equivalence” for Brownian motion Statistics B Probability Letters 1, 203-206 SINAi, JA G (1982) Limit behaviour of one-dimensional random walks in random environment Th of Probability and its Applications 27,247-258 SKOROHOD, A V (1961) Studies in the Theory of Random Processes Addison - Wesley, Reading, Mass SOLOMON, F (1975) Random walks in random environment The Annals of Probability 3, 1-31 SPITZER, F (1958) Some theorems concerning 2-dimensional Brownian motion Ransuctions of the A m Math SOC.87, 187-197 (1964) Principles of Random Walk Van Nostrand, Princeton, N.J STRASSEN, V (1964) An invariance principle for the law of iterated logarithm Wahrscheinhhkeitstheorae verw Gebiete 3, 211-226 (1966) A converse to the law of the iterated logarithm Wahrscheinlichkeitstheorie uerw Gebiete 4, 265-268 SZABADOS, T (1989) A discrete It6 formula Coll Math SOC J Bolyai: Limit Theorems in Probability and Statistics (ed I Berkes, E Csiiki, P RBvCsz) North-Holland 491-502 SZEKELY, G - TUSNADY, G (1979) Generalized Fibonacci numbers, and the number of “pure heads” Matematikai Lapok 27, 147-151 In Hungarian TOTH, B (1985) A lower bound for the critical probability of the square lattice site percolation Z Wahrscheinlichkeitstheorie verw Gebiete 69, 19-22 (1995) The (‘true’’ self-avoiding walk with bond repulsion on Z: limit theorems The Annals of Probability 23, 1523-1556 (1996/A) Multiple covering of the range of a random walk on Z (On a question of P Erdos and P RCvCsz) Studia Sci Math Hung 31, 355-359 REFERENCES 373 (1996/B) Generalized Ray - Knight theory and limit theorems for selfinteracting random walks on Z' The Annals of Probability 24, 1324-1367 (1997) Limit theorems for weakly reinforced random walks on Z Studia Sci Math Hung 33, 321-337 (1999) Self-interacting random motions - A survey Bolyai SOC Math Studies (ed: P RBvBsz, B T6th) 349-384 (2001) No more than three favorite sites for simple random walk The Annals of Probability 29, 484-503 TOTH, B -WERNER, W (1997) Tied favourite edges for simple random walk Combinatorics, Probability and Computing 6, 359-396 TROTTER, H F (1958) A property of Brownian motion paths Illinois J of Math 2, 425-433 WEIGL, A (1989) Zwei Satze uber die Belegungszeit beam Random Walk Diplomarbeit, TU Wien WICHURA, M (1977) Unpublished manuscript ZIMMERMANN, G (1972) Some sample function properties of the two-parameter Gaussian process Ann Math Statistics 43, 123551246, This page intentionally left blank Author Index Adelman, 279 Andjel, E D 348, 349 Auer, P 245, 251, 264, 266, 297 Donsker, M D 126, 224 Durett, R 14 Dvoretzky, A 209, 211, 221, 242 Bktfai, P 53, 77 Bass, R F 157, 222, 225 Benjamini, I 60 Berkes, I 32 Bickel, P J 180 Billingsley, P 16 Bingham, N H 38 Bolthausen, E 92, 351 Book, S A 71 Borel, E 28, 29 Borodin, A N 107, 110, 351 Brosamler, G A 140, 141 Burdzy, K 279 Erd&, P 17, 18, 34, 39, 53, 59, 62, 65, 67, 77, 79, 121, 135-139, 157, 158, 160, 180, 194, 209, 211, 213, 215, 219, 220, 221, 241, 242, 243, 245, 272, 278 Feller, W 19, 20, 34 Fisher, A 140 Flatto, L 220 Foldes, A 18, 21, 74, 81, 82, 107, 118123, 127, 134, 139, 155, 166, 167, 169, 172-175, 227, 236, 238 Gnedenko, B V 19 Golosov, A 345, 346, 347 Chen, R W 62 Goncharov, V L 21 Chen, X 243, 244 Goodman, V 94 Chung, K L 72, 112, 118, 121, 135, 136 Gorn, N L 94 Coppersmith, N 354 Griffin, P 157, 199, 233 CsBki, E 18, 41, 45, 71, 72, 74, 81, 82, Grill, K 18, 45, 68, 70, 71, 79, 92, 176, 88, 93, 94, 107, 112-115, 118178, 179 123, 125, 127, 137-139, 152, Guibas, L J 59, 62 159, 166, 167, 169, 172-176, 179, 227, 236, 238, 281, hSHHDYTP, 285, 351 hSMANA, y 220, 225, 235 cSORGO, m 66, 73, 110., 119, 120 126 HANSON, d l 72, 1911 141, 163, 164, 166, 167, 185, hART,AM p 53 17, H 318 hOVATH, l 110, 317, 318 Hough, J B 245 Darling, D A 180 Hu, Y 328, 334 Davis, B 354 Hunt, G A 112, 118 De Acosta, A 93 Deheuvels, P 18, 64, 75, 79, 80, 326, Imhof, p 175 327 ItB, K 141, 183 Dembo, A 219, 240, 246, 263, 277 Devroye, L 80 Jain, N C 222, 225 Diaconis, P 354 Kakutani, S 242 Dobrushin, R L 129, 130 375 376 AUTHOR INDEX Pruitt, W E 208, 222, 225 Kalikow, S A 349 Puri, M L 155 Karlin, S 66 Kesten, H 112, 118, 296-298, 346, 350, 351 Qualls, G 180, 182 Key, E S 347, 348 RCnyi, A 13, 14, 19, 20, 28, 53, 67, 77, Khinchine, A 30 101 Khoshnevisan, D 107, 108, 146, 164 RCvCsz, P 17, 18, 39, 59, 63, 65, 66, 71Knight, F B 52, 111, 139, 204, 205 73, 75, 79, 88, 91, 109, 112-115, Kolmogorov, A N 19, 25, 34, 38 119, 120, 125, 126, 134, 137, Komlos, J 18, 53, 54 138, 141, 157-160, 163, 164, Kuelbs, J 94 166, 167, 172-175, 185, 227, Kumagai, T 222, 225 235, 236, 238, 245, 246, 256, Kusolitsch, N 66 263, 264, 266, 269, 272, 278, 281, 283, 285, 317, 318, 326, Lacey, M T 141 327, 337, 351 Lamperti, J 293 Riesz, F 85 Lawler, G F 195, 242, 244, 246 Rosen, J 219, 236, 238, 240, 244, 246, Le Gall, J F 225, 226 263, 281, 283 LCvy, P 33, 104, 111, 140, 141 Rosenblatt, M 180 Li, W 243 Russo, R P 72, 181 Lifschitz, M A 94 Lynch, I 80 Sarnarova, S S 59 Major, P 53, 54, 161 Marcus, M B 244 McKean Jr, H P 141, 185 Mogul’skii, A A 116 Mori, T 62, 63 Mueller, C 95, 126 Nemetz, T 66 Newrnan, D 161 Odlyzko, A M 59, 62 Orey, S 208 Ortega, I 68, 72 Ost, F 66 Pemantle, R 279 Peres, Y 60, 219, 240, 245, 246, 263 Perkins, E 119, 141, 240 Petrov, V V 66 Petrowsky, I G 34 Philipp, W 141 Pitt, J H 220 Polya, G 23, 193 Schatte, P 140 Shao, Q M 182 Shi, Z 158, 159, 236, 238, 263, 283, 285, 328, 334, 351 Shore, T R 71 Simons, G 113, 115 Sinai, JA, G 314, 345, 346 Skorohod, A V 53 Solomon, F 311 Spitzer, F 27, 205, 206, 248, 350, 351 Steinebach, J 80 Steif, J E 60 Strassen, V 32, 88, 89 Szabados, T 183 SzCkely, G 18 Sz.-Nagy, B 85 Taylor, S J 139, 194, 209, 211, 213, 215, 219, 220, 240, 241, 242, 243 T6th, B 158, 159, 162, 297, 355 Trotter, H F 105 TusnBdy, G 18, 53, 54 Varadhan, S R 126, 224 A UTHOR INDEX Vincze, I 113 Watanabe, H 180, 183 Weigl, A 141 Werner, W 159 Wichura, M 95, 126 377 Wintner, A 53 Wschebor, M 68, 72 Zeitouni, 219, 240, 246, 263 Zimmermann, G 163 This page intentionally left blank Subject Index Arcsine law 104 Asymptotically deterministic sequence 34 Bernstein inequality 13 Bore1 - Cantelli lemma 27 Brownian motion Central limit theorem 19 Chebyshev inequality 28 Dirichlet problem 293 DLA model 296 EFKP LIL 34 Gap method 29 Invariance principle 52, 109, 203 It6 formula 183 It8 integral 183 Large deviation theorem 14, 19 Levy classes 33 LIL of Hartman - Wintner 32 LIL of Khinchine 31 Logarithmic density 140 Long head-runs 57 Markov inequality 28 Method of high moments 29 Normal numbers 29 Ornstein - Uhlenbeck process 179 Percolation 297 Quasi asymptotically deterministic sequence 34 Rademacher functions 10, 11 Random walk in random environment definition 303, 348 local time 313, 330, 335, 337 maximum 313, 325, 327, 345 recurrence 311 Random walk in random scenery 350 ' Random walk in Z definition excursion 146, 147 favourite sites 157 first recurrence 97 increments 57, 77 increments of the local time 123 law of the iterated logarithm 31 law of the large numbers 28 local time 98, 117, 146, location of the last zero 102, 136, 175 location of the maximum 102, 104, 136, 171 longest run 21, 57 longest zero-free interval 135 maximum 14, 20, 31, 35, 41 maximum of the absolute value 20, 31, 35, 41, 171 mesure du voisinage 141 number of crossings 100, 113 range 44 rarely visited points 157, 161 recurrence 23 Strassen type theorems 83, 90, 124 Random walk in Z d almost covered discs 264 completely covered balls 272 completely covered discs 245, 263 definition 192 excursions 281, 284 first recurrence 211 heavy points 227 heavy balls 236 law of the iterated logarithm 207 379 380 local time 211, 218 maximum 206, 209 range 221 rate of escape 209 recurrence 193 self-crossing 241 speed of escape 288 Strassen type theorems 204 Reflection principle 15 Reinforced random walk 353 SUBJECT I N D E X excursion 139, 141 increments 66 increments of the local time 119 local time 104, 109, location of the last zero 179 location of the maximum 175 longest zero-free interval 121 maximum 53 maximum of the absolute value 53 mesure du voisinage 141 occupation time 155 Skorohod embedding scheme 53 Strassen type theorems 84, 90, 92, Stirling formula 19 124 Wiener process in Tanaka formula 185 definition 203 Theorem of Bore1 29 law of the iterated logarithm 207 Theorem of Chung 39 maximum 206 Theorem of Donsker and Varadhan 125 rate of escape 209 Theorem of Hausdorff 30 self-crossing 241 Theorem of Hirsch 39 Strassen type theorems 204 Wiener sausage 226 Wichura's theorem 95 Wiener sheet 163 Wiener process in R' definition 51 Zero-one law 25 ... N O N- RAND O M ENVIRONMENTS h E C D N D E D I T I O N This page intentionally left blank L RANDEOM WALK IN RANDOM AND N NON RANDOM N M ENVIRONMENTS ENVIRO E C O N EDITION D Pal Revesz Technische... presented In some cases more proofs are given, in some cases none The proofs are omitted when they can be obtained by routine methods and when they are too long and too technical In both cases... interesting ones and to fit them in this Second Edition Many of my friends helped me to find the most important new results and to discover some of the mistakes in the First Edition My special thanks