Stopped random walks limit theorems and applications

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Springer Series in Operations Research and Financial Engineering Series Editors: Thomas V Mikosch Sidney I Resnick Stephen M Robinson For other titles published in this series, go to http://www.springer.com/series/3182 www.ebook3000.com Allan Gut Stopped Random Walks Limit Theorems and Applications Second Edition 123 Allan Gut Department of Mathematics Uppsala University SE-751 06 Uppsala Sweden allan.gut@math.uu.se Series Editors: Thomas V Mikosch University of Copenhagen Laboratory of Actuarial Mathematics DK-1017 Copenhagen Denmark mikosch@act.ku.dk Stephen M Robinson University of Wisconsin-Madison Department of Industrial Engineering Madison, WI 53706 USA smrobins@facstaff.wise.edu Sidney I Resnick Cornell University School of Operations Research and Industrial Engineering Ithaca, NY 14853 USA sirl@cornell.edu ISSN 1431-8598 ISBN 978-0-387-87834-8 DOI 10.1007/978-0-387-87835-5 e-ISBN 978-0-387-87835-5 Library of Congress Control Number: 2008942432 Mathematics Subject Classification (2000): 60G50, 60K05, 60F05, 60F15, 60F17, 60G40, 60G42 c Springer Science+Business Media, LLC 1988, 2009 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form 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 terms, 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 rights Printed on acid-free paper springer.com www.ebook3000.com Preface to the 1st edition My first encounter with renewal theory and its extensions was in 1967/68 when I took a course in probability theory and stochastic processes, where the then recent book Stochastic Processes by Professor N.U Prabhu was one of the requirements Later, my teacher, Professor Carl-Gustav Esseen, gave me some problems in this area for a possible thesis, the result of which was Gut (1974a) Over the years I have, on and off, continued research in this field During this time it has become clear that many limit theorems can be obtained with the aid of limit theorems for random walks indexed by families of positive, integer valued random variables, typically by families of stopping times During the spring semester of 1984 Professor Prabhu visited Uppsala and very soon got me started on a book focusing on this aspect I wish to thank him for getting me into this project, for his advice and suggestions, as well as his kindness and hospitality during my stay at Cornell in the spring of 1985 Throughout the writing of this book I have had immense help and support from Svante Janson He has not only read, but scrutinized, every word and every formula of this and earlier versions of the manuscript My gratitude to him for all the errors he found, for his perspicacious suggestions and remarks and, above all, for what his unusual personal as well as scientific generosity has meant to me cannot be expressed in words It is also a pleasure to thank Ingrid Torr˚ ang for checking most of the manuscript, and for several discoveries and remarks Inez Hjelm has typed and retyped the manuscript My heartfelt thanks and admiration go to her for how she has made typing into an art and for the everlasting patience and friendliness with which she has done so The writing of a book has its ups and downs My final thanks are to all of you who shared the ups and endured the downs Uppsala September 1987 Allan Gut Preface to the 2nd edition By now Stopped Random Walks has been out of print for a number of years Although 20 years old it is still a fairly complete account of the basics in renewal theory and its ramifications, in particular first passage times of random walks Behind all of this lies the theory of sums of a random number of (i.i.d.) random variables, that is, of stopped random walks I was therefore very happy when I received an email in which I was asked whether I would be interested in a reprint, or, rather, an updated 2nd edition of the book And here it is! To the old book I have added another chapter, Chapter 6, briefly traversing nonlinear renewal processes in order to present more thoroughly the analogous theory for perturbed random walks, which are modeled as a random walk plus “noise”, and thus behave, roughly speaking, as O(n)+o(n) The classical limit theorems as well as moment considerations are proved and discussed in this setting Corresponding results are also presented for the special case when the perturbed random walk on average behaves as a continuous function of the arithmetic mean of an i.i.d sequence of random variables, the point being that this setting is most apt for applications to exponential families, as will be demonstrated A short outlook on further results, extensions and generalizations is given toward the end of the chapter A list of additional references, some of which had been overlooked in the first edition and some that appeared after the 1988 printing, is also included, whether explicitly cited in the text or not Finally, many thanks to Thomas Mikosch for triggering me into this and for a thorough reading of the second to last version of Chapter Uppsala October 2008 Allan Gut www.ebook3000.com Contents Preface to the 1st edition v Preface to the 2nd edition vii Notation and Symbols xiii Introduction Limit Theorems for Stopped Random Walks 1.1 Introduction 1.2 a.s Convergence and Convergence in Probability 1.3 Anscombe’s Theorem 1.4 Moment Convergence in the Strong Law and the Central Limit Theorem 1.5 Moment Inequalities 1.6 Uniform Integrability 1.7 Moment Convergence 1.8 The Stopping Summand 1.9 The Law of the Iterated Logarithm 1.10 Complete Convergence and Convergence Rates 1.11 Problems 9 12 16 18 21 30 39 42 44 45 47 Renewal Processes and Random Walks 2.1 Introduction 2.2 Renewal Processes; Introductory Examples 2.3 Renewal Processes; Definition and General Facts 2.4 Renewal Theorems 2.5 Limit Theorems 2.6 The Residual Lifetime 2.7 Further Results 2.7.1 49 49 50 51 54 57 61 64 64 x Contents 2.8 2.9 2.10 2.11 2.12 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 Random Walks; Introduction and Classifications Ladder Variables The Maximum and the Minimum of a Random Walk Representation Formulas for the Maximum Limit Theorems for the Maximum 65 65 66 66 66 66 69 71 72 74 Renewal Theory for Random Walks with Positive Drift 79 3.1 Introduction 79 3.2 Ladder Variables 82 3.3 Finiteness of Moments 83 3.4 The Strong Law of Large Numbers 88 3.5 The Central Limit Theorem 91 3.6 Renewal Theorems 93 3.7 Uniform Integrability 96 3.8 Moment Convergence 98 3.9 Further Results on Eν(t) and Var ν(t) 100 3.10 The Overshoot 103 3.11 The Law of the Iterated Logarithm 108 3.12 Complete Convergence and Convergence Rates 109 3.13 Applications to the Simple Random Walk 109 3.14 Extensions to the Non-I.I.D Case 112 3.15 Problems 112 Generalizations and Extensions 115 4.1 Introduction 115 4.2 A Stopped Two-Dimensional Random Walk 116 4.3 Some Applications 126 4.3.1 Chromatographic Methods 126 4.3.2 Motion of Water in a River 129 4.3.3 The Alternating Renewal Process 129 4.3.4 Cryptomachines 130 4.3.5 Age Replacement Policies 130 4.3.6 Age Replacement Policies; Cost Considerations 132 4.3.7 Random Replacement Policies 132 4.3.8 Counter Models 132 4.3.9 Insurance Risk Theory 133 4.3.10 The Queueing System M/G/1 134 4.3.11 The Waiting Time in a Roulette Game 134 4.3.12 A Curious (?) Problem 136 4.4 The Maximum of a Random Walk with Positive Drift 136 www.ebook3000.com Contents xi 4.5 First Passage Times Across General Boundaries 141 Functional Limit Theorems 157 5.1 Introduction 157 5.2 An Anscombe–Donsker Invariance Principle 157 5.3 First Passage Times for Random Walks with Positive Drift 162 5.4 A Stopped Two-Dimensional Random Walk 167 5.5 The Maximum of a Random Walk with Positive Drift 169 5.6 First Passage Times Across General Boundaries 170 5.7 The Law of the Iterated Logarithm 172 5.8 Further Results 174 Perturbed Random Walks 175 6.1 Introduction 175 6.2 Limit Theorems; the General Case 178 6.3 Limit Theorems; the Case Zn = n · g(Y¯n ) 183 6.4 Convergence Rates 190 6.5 Finiteness of Moments; the General Case 190 6.6 Finiteness of Moments; the Case Zn = n · g(Y¯n ) 194 6.7 Moment Convergence; the General Case 198 6.8 Moment Convergence; the Case Zn = n · g(Y¯n ) 200 6.9 Examples 202 6.10 Stopped Two-Dimensional Perturbed Random Walks 205 6.11 The Case Zn = n · g(Y¯n ) 209 6.12 An Application 211 6.13 Remarks on Further Results and Extensions 214 6.14 Problems 221 A Some Facts from Probability Theory 223 A.1 Convergence of Moments Uniform Integrability 223 A.2 Moment Inequalities for Martingales 225 A.3 Convergence of Probability Measures 229 A.4 Strong Invariance Principles 234 A.5 Problems 235 B Some Facts about Regularly Varying Functions 237 B.1 Introduction and Definitions 237 B.2 Some Results 238 References 241 Index 257 Notation and Symbols x∨y x∧y x+ x− [x] I{A} Card{A} d X =Y a.s Xn −−→ X p Xn − →X d →X Xn − =⇒ J =⇒ M1 =⇒ σ{Xk , ≤ k ≤ n} EX exists X r Φ(x) W (t) i.i.d i.o iff max{x, y} min{x, y} x∨0 −(x ∧ 0) the largest integer in x, the integral part of x the indicator function of the set A the number of elements in the set A X and Y are equidistributed Xn converges almost surely to X Xn converges in probability to X Xn converges in distribution to X weak convergence weak convergence in the Skorohod J1 -topology weak convergence in the Skorohod M1 -topology the σ-algebra generated by X1 , X2 , , Xn at least one of EX − and EX + are finite (E|X|r )1/r x √ e−y /2 dy (−∞ < x < ∞) 2π −∞ Brownian motion or the Wiener process independent, identically distributed infinitely often if and only if end of proof www.ebook3000.com Introduction A random walk is a sequence {Sn , n ≥ 0} of random variables with independent, identically distributed (i.i.d.) increments {Xk , k ≥ 1} and S0 = A Bernoulli random walk (also called a Binomial random walk or a Binomial process) is a random walk for which the steps equal or with probabilities p and q, respectively, where < p < and p + q = A simple random walk is a random walk for which the steps equal +1 or −1 with probabilities p and q, respectively, where, again, < p < and p + q = The case p = q = 12 is called the symmetric simple random walk (sometimes the coin-tossing random walk or the symmetric Bernoulli random walk) A renewal process is a random walk with nonnegative increments; the Bernoulli random walk is an example of a renewal process Among the oldest results for random walks are perhaps the Bernoulli law of large numbers and the De Moivre–Laplace central limit theorem for Bernoulli random walks and simple random walks, which provide information about the asymptotic behavior of such random walks Similarly, limit theorems such as the classical law of large numbers, the central limit theorem and the Hartman– Wintner law of the iterated logarithm can be interpreted as results on the asymptotic behavior of (general) random walks These limit theorems all provide information about the random walks after a fixed number of steps It is, however, from the point of view of applications, more natural to consider random walks evaluated at fixed or specific random times, and, hence, after a random number of steps Namely, suppose we have some application in mind, which is modeled by a random walk; such applications are abundant Let us just mention sequential analysis, queueing theory, insurance risk theory, reliability theory and the theory of counters In all these cases one naturally studies the process (evolution) as time goes by In particular, it is more interesting to observe the process at the time point when some “special” event occurs, such as the first time the process exceeds a given value rather than the time points when “ordinary” events occur From this point of view it is thus more relevant to study randomly indexed random walks 248 References 156 Hartman, P and Wintner, A (1941): On the law of the iterated logarithm Amer J Math 63, 169–176 157 Hat¯ ori, H (1959): Some theorems in an extended renewal theory I K¯ odai Math Sem Rep 11, 139–146 158 Heyde, C.C (1964): Two probability theorems and their applications to some first passage problems J Austral Math Soc 4, 214–222 159 Heyde, C.C (1966): Some renewal theorems with applications to a first passage problem Ann Math Statist 37, 699–710 160 Heyde, C.C (1967a): Asymptotic renewal results for a natural generalization of classical renewal theory J Roy Statist Soc Ser B 29, 141–150 161 Heyde, C.C (1967b): A limit theorem for random walks with drift J Appl Probab 4, 144–150 162 Hinderer, K and Walk, H (1974): A generalization of renewal processes In: Progress in Statistics (European Meeting of Statisticians, Budapest, 1972), Vol I, 315–318 Colloq Math Soc Janos Bolyai, Vol 9, North-Holland, Amsterdam 163 Hă ogfeldt, P (1977): On the asymptotic behaviour of first passage time processes and certain stopped stochastic processes Abstract In: Proc 2nd Vilnius Conf on Prob and Math Stat Vol 3, 86–87 164 Horv´ ath, L (1984a): Strong approximation of renewal processes Stoch Process Appl 18, 127–138 165 Horv´ ath, L (1984b): Strong approximation of extended renewal processes Ann Probab 12, 1149–1166 166 Horv´ ath, L (1984c): Strong approximation of certain stopped sums Statist Probab Lett 2, 181–185 167 Horv´ ath, L (1985): A strong nonlinear renewal theorem with applications to sequential analysis Scand J Statist 12, 271–280 168 Horv´ ath, L.(1986): Strong approximations of renewal processes and their applications Acta Math Acad Sci Hungar 47, 13–28 169 Hsu, P.L and Robbins, H (1947): Complete convergence and the law of large numbers Proc Nat Acad Sci U.S.A 33, 25–31 170 Hu, I (1988): Repeated significance tests for exponential families Ann Statist 16, 1643–1666 171 Hu, I (1991): Nonlinear renewal theory for conditional random walks Ann Probab 19, 401–422 172 Huggins, R.M (1985): Laws of the iterated logarithm for time changed Brownian motion with an application to branching processes Ann Probab 13, 1148–1156 173 Hunter, J.J (1974a): Renewal theory in two dimensions: Basic results Adv in Appl Probab 6, 376–391 174 Hunter, J.J (1974b): Renewal theory in two dimensions: Asymptotic results Adv in Appl Probab 6, 546–562 175 Hunter, J.J (1977): Renewal theory in two dimensions: Bounds on the renewal function Adv in Appl Probab 9, 527–541 176 Iglehart, D.L and Whitt, W (1971): The equivalence of functional central limit theorems for counting processes and associated partial sums Ann Math Statist 42, 1372–1378 177 Jagers, P (1975): Branching Processes with Biological Applications John Wiley, New York References 249 178 Janson, S (1983): Renewal theory for m-dependent variables Ann Probab 11, 558–568 179 Janson, S (1986): Moments for first passage and last exit times, the minimum and related quantities for random walks with positive drift Adv in Appl Probab 18, 865–879 180 Kaijser, T (1971): A stochastic model describing the water motion in a river Nordic Hydrology II, 243–265 181 Karamata, J (1930): Sur une mode de croissance r´eguli`ere des fonctions Mathematica (Cluj) 4, 38–53 182 Katz, M.L (1963): The probability in the tail of a distribution Ann Math Statist 34, 312–318 183 Keener, R (1987): Asymptotic expansions in nonlinear renewal theory In: New Perspectives in Theoretical and Applied Statistics (Bilbao, 1986), 479–502 Wiley Ser Probab Math Statist Probab Math Statist., Wiley, New York 184 Kemperman, J.H.B (1961): The Passage Problem for a Stationary Markov Chain University of Chicago Press, Chicago, IL 185 Kesten, H (1974): Renewal theory for functionals of a Markov chain with general state space Ann Probab 2, 355–386 186 Kiefer, J and Wolfowitz, J (1956): On the characteristics of the general queueing process, with applications to random walk Ann Math Statist 27, 147–161 187 Kingman, J.F.C (1968): The ergodic theory of subadditive stochastic processes J Roy Statist Soc Ser B 30, 499–510 188 Klesov, O., Rychlik, Z and Steinebach, J (2001): Strong limit theorems for general renewal processes Probab Math Statist 21, 329–349 189 Kolmogorov, A.N (1936): Anfangsgră unde der Theorie der Markoschen Ketten mit unendlich vielen mă oglichen Zustă anden Mat Sb (N.S.) 1, 607–610 190 Koml´ os, J., Major, P and Tusn´ ady, G (1975): An approximation of partial sums of independent r.v.’s and the sample d.f., I Z Wahrsch Verw Gebiete 32, 111–131 191 Koml´ os, J., Major, P and Tusn´ ady, G (1976): An approximation of partial sums of independent r.v.’s and the sample d.f., II Z Wahrsch Verw Gebiete 34, 33–58 192 Lai, T.L (1975): On uniform integrability in renewal theory Bull Inst Math Acad Sinica 3, 99–105 193 Lai, T.L (1976): Asymptotic moments of random walks with applications to ladder variables and renewal theory Ann Probab 4, 51–66 194 Lai, T.L (1977): First exit times from moving boundaries for sums of independent random variables Ann Probab 5, 210–221 195 Lai, T.L (1988): Boundary crossing problems for sample means Ann Probab 16, 375–396 196 Lai, T.L and Siegmund, D (1977): A nonlinear renewal theory with applications to sequential analysis Ann Statist 5, 946–954 197 Lai, T.L and Siegmund, D (1979): A nonlinear renewal theory with applications to sequential analysis II Ann Statist 7, 60–76 198 Lalley, S.P (1982): Nonlinear renewal theory for lattice random walks Comm Statist — Sequential Analysis 1, 193–205 199 Lalley, S.P (1983): Repeated likelihood ratio tests for curved exponential families Z Wahrsch verw Gebiete 62, 293–321 200 Lalley, S.P (1984a): Limit theorems for first-passage times in linear and nonlinear renewal theory Adv in Appl Probab 16, 766–803 www.ebook3000.com 250 References 201 Lalley, S.P (1984b): Conditional Markov renewal theory I Finite and denumerable state space Ann Probab 12, 1113–1148 202 Lalley, S.P (1986): Renewal theorem for a class of stationary sequences Probab Th Rel Fields 72, 195–213 203 Lamperti, J (1958): Some limit theorems for stochastic processes J Math Mech 7, 433–448 204 Lamperti, J (1961): A contribution to renewal theory Proc Amer Math Soc 12, 724–731 205 Larsson-Cohn, L (1999): Some Limit and Continuity Theorems for Perturbed Random Walks Licentiat thesis Report U.U.D.M 1999:2, Uppsala University, Sweden 206 Larsson-Cohn, L (2000a): Invariance priniples for the first passage times of perturbed random walks Statist Probab Lett 48, 347–351 207 Larsson-Cohn, L (2000b): A global approach to first passage times Probab Math Statist 20, 337–341 208 Larsson-Cohn, L (2001a): Some remarks on “First passage times of perturbed random walks” Sequential Anal 20, 87–90 209 Larsson-Cohn, L (2001b): On the difference between two first passage times Studia Sci Math Hung 37, 53–67 210 Lindberger, K (1978): Functional limit theorems for cumulative processes and stopping times Z Wahrsch verw Gebiete 44, 47–56 211 Lindvall, T (1973): Weak convergence of probability measures and random functions in the function space D[0, ∞) J Appl Probab 10, 109–121 212 Lindvall, T (1977): A probabilistic proof of Blackwell’s renewal theorem Ann Probab 5, 482–485 213 Lindvall, T (1979): On coupling of discrete renewal processes Z Wahrsch verw Gebiete 48, 57–70 214 Lindvall, T (1982): On coupling of continuous-time renewal processes J Appl Probab 19, 82–89 215 Lindvall, T (1986): On coupling of renewal processes with use of failure rates Stoch Process Appl 22, 1–15 216 Lo`eve, M (1977): Probability Theory, 4th ed Springer-Verlag, New York 217 Lorden, G (1970): On excess over the boundary Ann Math Statist 41, 520–527 218 Maejima, M (1975): On local limit theorems and Blackwell’s renewal theorem for independent random variables Ann Inst Statist Math 27, 507–520 219 Maejima, M and Mori, T (1984): Some renewal theorems for random walks in multidimensional time Math Proc Cambridge Philos Soc 95, 149–154 220 Marcinkiewicz, J and Zygmund, A (1937): Sur les fonctions ind´ependantes Fund Math 29, 60–90 221 Marcinkiewicz, J and Zygmund, A (1938): Quelques th´eor`emes sur les fonctions ind´ependantes Studia Math VII, 104–120 222 Melfi, V.F (1992): Nonlinear Markov renewal theory with statistical applications Ann Probab 20, 753–771 223 Melfi, V.F (1994): Nonlinear renewal theory for Markov random walks Stoch Process Appl 54, 71–93 224 Meyer, P.A (1966): Probability and Potential Waltham, Blaisdell 225 Mikosch, T (2006): Non-Life Insurance Mathematics An Introduction with Stochastic Processes, Corr 2nd printing Springer-Verlag, Berlin References 251 226 Mohan, N.R (1976): Teugels’ renewal theorem and stable laws Ann Probab 4, 863–868 227 Nagaev, A.V (1979): Renewal theorems in Rd Theory Probab Appl XXIV, 572–581 228 Neveu, J (1975): Discrete-Parameter Martingales North-Holland, Amsterdam 229 Ney, P (1981): A refinement of the coupling method in renewal theory Stoch Process Appl 11, 11–26 230 Ney, P and Wainger, S (1972): The renewal theorem for a random walk in two-dimensional time Studia Math XLIV, 71–85 231 Niculescu, S ¸ P (1979): Extension of a renewal theorem Bull Math Soc Sci Math R.S Roumanie 23, 289–292 232 Ornstein, D.S (1969a): Random walks I Trans Amer Math Soc 138, 1–43 233 Ornstein, D.S (1969b): Random walks II Trans Amer Math Soc 138, 45–60 234 Peligrad, M (1999): Convergence of stopped sums of weakly dependent random variables Electronic J Probab 4, 1–13 235 Pluci´ nska, A 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atze in der Nichtlinearen Erneuerungstheorie und Anwendungen in der Sequentialanalysis Dissertation, University of Stuttgart Mitt Math Sem Giessen 172, iv + 98 pp 294 Su, Y.T (1993): Limit theorems for first-passage times in nonlinear Markov renewal theory Sequential Anal 12, 235–245 295 Su, Y.T (1998): A renewal theory for perturbed Markov random walks with applications to sequential analysis Statistica Sinica 8, 429–443 296 Su, Y.T and Wang, C.T (1998): Nonlinear renewal theory for lattice Markov random walk Sequential Anal 17, 267–277 297 Szynal, D (1972): On almost complete convergence for the sum of a random number of independent random variables Bull Acad Polon Sci S´er Sci Math Astronom Phys 20, 571574 www.ebook3000.com 254 References 298 Tă acklind, S (1944): Elementare Behandlung vom Erneuerungsproblem fă ur den stationă aren Fall Skand Aktuarietidskr 27, 1–15 299 Tak´ acs, L (1956): On a probability problem arising in the theory of counters Proc Cambridge Philos Soc 52, 488–498 300 Tak´ acs, L (1957): On certain sojourn time problems in the theory of stochastic processes Acta Math Acad Sci Hungar 8, 169–191 301 Takahashi, H and Woodroofe, M (1981): Asymptotic expansions in nonlinear renewal theory Comm Statist A—Theory Methods 10, 2113–2135 302 Teicher, H (1973): A classical limit theorem without invariance or reflection Ann Probab 1, 702–704 303 Teugels, J.L (1968): Renewal theorems when the first or the second moment is infinite Ann Math Statist 39, 1210–1219 304 Thorisson, H (1987): A complete coupling proof of Blackwell’s renewal theorem Stoch Process Appl 26, 87–97 305 Torrang, ˙ I (1987): The law of the iterated logarithm—cluster points of deterministic and random subsequences Probab Math Statist VIII, 133–141 306 Vervaat, W (1972a): Success Epochs in Bernoulli Trials (with Applications in Number Theory) Mathematical Centre Tracts 42, Amsterdam 307 Vervaat, W (1972b): Functional central limit theorems for processes with positive drift and their inverses Z Wahrsch verw Gebiete 23, 245–253 308 von Bahr, B (1974): Ruin probabilities expressed in terms of ladder height distributions Scand Actuar J., 190–204 309 Wald, A (1947): Limit distribution of the maximum and minimum of successive cumulative sums of random variables Bull Amer Math Soc 53, 142–153 310 Walk, H (1989): Nonlinear renewal theory under growth conditions Stoch Process Appl 32, 289–303 311 Wang, M (1992): A local limit theorem for perturbed random walks Statist Probab Lett 15, 109–116 312 Wang, M (1996): Limit theorems on occupation times for perturbed random walks Sequential Anal 15, 103–122 313 Whitt, W (1980): Some useful functions for functional limit theorems Math Oper Res 5, 67–81 314 Wijsman, R.A (1991): Stopping times: Termination, moments, distribution In: Handbook of Sequential Analysis (Eds B.K Ghosh and P.K Sen), 67–119 Statist Textbooks Monogr., 118, Marcel Dekker, New York 315 Williamson, J.A (1965): Some renewal theorems for non-negative independent random variables Trans Amer Math Soc 114, 417–445 316 Williamson, J.A (1968): Random walks and Riesz kernels Pacific J Math 25, 393–415 317 Woodroofe, M (1976): A renewal theorem for curved boundaries and moments of first passage times Ann Probab 4, 67–80 318 Woodroofe, M (1978): Large deviations of likelihood ratio statistics with applications to sequential testing Ann Statist 6, 72–84 319 Woodroofe, M (1982): Nonlinear Renewal Theory in Sequential Analysis CBMS–NSF Regional Conf Ser in Appl Math 39 SIAM, Philadelphia, PA 320 Woodroofe, M and Keener, R (1987): Asymptotic expansions in boundary crossing problems Ann Probab 15, 102–114 321 Woodroofe, M (1988): Asymptotic expansions for first passage times Stoch Process Appl 28, 301–315 References 255 322 Yu, K.F (1979): On the uniform integrability of the normalized randomly stopped sums of independent random variables Preprint, Yale University 323 Zhang, C.-H (1988): A nonlinear renewal theory Ann Probab 16, 793–824 324 Zhang, C.-H (1989): A renewal theory with varying drift Ann Probab 17, 723–736 www.ebook3000.com Index Ahlberg, 7, 126 Ale˘skeviˇcience, 92 Alsmeyer, 214, 216, 219 alternative hypothesis, 211 Anderson, 64 Anscombe, 3, 4, 9, 16, 181 condition, 16, 47, 176, 179, 180, 207, 215 Anscombe’s theorem, 3, 16, 18, 35, 39, 47, 57, 59, 91, 146, 157, 160, 162 Lr -analogue of, 35 multidimensional, 160 Anscombe–Donsker invariance principle, 157 Arjas, 56 Asmussen, 4, 51, 66, 116, 133 Athreya, 56, 64 Barlow, 130–132 Basu, 162 Baum, 45, 190 Berbee, 112 Berry, 65, 218 Bickel, 66 Billingsley, 7, 20, 125, 159, 160, 162, 163, 224, 229, 233 Bingham, 109, 166, 235 binomial process, Blackwell, 9, 25, 29, 47, 55, 64, 81, 93–96 Breiman, Brown, Brownian motion, 157, 173, 235 Burkholder, 22, 31, 198, 225, 227, 228 (C, C ), 158, 229, 230 Carlsson, 66, 95, 105 central limit theorem, 1, 7, 9, 16, 18, 20, 32, 39, 46, 57, 59, 60, 65, 76, 77, 80, 81, 91, 92, 97, 98, 108, 119, 139, 146, 147, 150, 153, 157, 178, 179, 184, 209, 210, 225–227, 234 moment convergence, 18, 115 multidimensional, 125 Chang, 35, 173 Choquet, 56 Chow, 7, 18, 25, 26, 35, 36, 71, 98, 99, 101, 108, 112, 140, 147, 151, 152, 154, 155 chromatography, 6, 126, 127 Chung, 4, 16, 66, 69, 77, 95, 105, 223, 224, 226, 228 C ¸ inlar, 4, 51, 66 cluster set, 44, 235 combinatorial methods, 4, 110 composition, 161, 166, 233 continuity of function(al)s, 167, 177, 233 composition, 161, 233 first passage time, 175, 176 inversion, 80 largest jump, 164 projections, 160, 166 supremum, 165 continuous mapping theorem, 159, 160, 164, 231, 232 convergence, 258 Index almost sure (a.s.), 6, 10, 12–14, 18, 58, 82, 187, 224, 235 complete, 10, 45, 46, 109, 125 in Lr , 4, 10, 19, 224 in r-mean, in distribution, 4, 10, 16, 18, 39, 59, 72, 77, 223, 224, 229, 232, 236 in probability, 4, 10, 12, 17, 158, 223, 224, 234 moment, 4, 10, 18, 19, 39, 41, 57, 60, 81, 82, 98, 104, 115, 140, 153, 223–226 of finite-dimensional distributions, 160, 161, 166, 229, 230 rate, 45, 109, 178, 190 convergence of probability measures, 229 convergence rates, 109, 190 convex conjugate, 204, 212 counters, 1, 2, 132, 133 counting process, 2, 3, 9, 10, 51, 52, 54, 57, 65, 72, 80, 82, 87, 88, 91, 137, 219, 220 coupling, 56 Cox, 112, 116, 130 Cramer, 133 Csă org o, 217, 218 (D, D), 230, 233 D0 , 165, 233 Daley, 95 Davis, 22, 227 De Acosta, 235 De Groot, 25 de Haan, 143, 237 Deny, 56 difference between two stopping times, 214 Doeblin, 56 domain of attraction, 18 Doney, 66, 71 Donsker, 7, 157, 217 Donsker’s theorem, 157–159, 168, 172, 174, 230, 234, 235 Anscombe version of, 172 Doob, 22, 51, 56, 57, 93, 225, 226, 228 Dynkin, 65 Edgeworth expansions, 218 Englund, 65 Erd˝ os, 6, 7, 45, 55, 77, 216 Erd˝ os-R´enyi laws, 216 Erickson, 57, 64, 65 Ess´en, 65, 95 Esseen, 218 exponential families, 178, 203, 205, 209 Farrell, 66 Feller, 4, 6, 26, 39, 51, 55, 57, 64, 65, 69, 92, 96, 123, 143, 237 Fenchel, 204 finiteness of moments, 21, 22, 24, 26–28, 83, 84, 86, 87, 96, 142, 190, 194 first passage time(s), 2, 5–7, 9, 10, 23, 28, 42, 54, 80, 89, 112, 116, 136, 140, 142, 154, 157, 160, 162, 165, 173, 176, 206, 218 across general boundaries, 2, 81, 141, 170, 173 auxiliary, 148 central limit theorem, 91, 115 complete convergence, 109 convergence rate, 109 excess over the boundary, 103 for the ladder height process, 82 law of large numbers, 1, 68, 75, 88, 138 law of the iterated logarithm, 81, 108, 115, 140, 155, 157 moment generating function, 87, 143 moments convergence, 97, 104 finiteness, 83, 96, 142 overshoot, 81, 103 process, 3, 5, 6, 54, 58, 60, 80, 127, 147, 165 subadditivity, 58, 110 uniform integrability, 60, 82, 96, 147 weak convergence, 168, 229 fluctuation theory, 4, 49 Fuchs, 4, 69 functional central limit theorem, 157, 230 limit theorem, 7, 162, 174 Gafurov, 66 Garsia, 64, 225 generalized arc sine distribution, 65 www.ebook3000.com Index Gikhman, 161, 231 Gnedenko, 130 Goldie, 109 Gră ubel, 96 Gundy, Gut, 6, 7, 23, 28, 30, 37, 38, 41, 42, 45, 47, 80, 81, 83, 89, 91, 92, 99–102, 104–106, 108, 109, 112, 116, 119, 120, 122, 124, 126, 130, 134, 135, 142, 143, 145–149, 154–156, 160, 162, 163, 166, 168, 169, 171, 239 Hall, 173 Harris recurrence, 219 Hartman, 44, 234, 235 Hat¯ ori, 57 Heath, 136 Heyde, 77, 83, 86–88, 91, 92, 94, 101, 140, 173 Hă ogfeldt, 168 Holst, 134, 135 Horv´ ath, 173, 218 Hsiung, 18, 35, 36, 98, 99, 108, 140, 147, 151, 152, 154, 155, 173 Hsu, 45 Huggins, 173 Hunter, 66 Iglehart, 166, 174 insurance risk theory, 1, 2, 66, 133 invariance principle, 157, 232 almost sure, 235 Anscombe–Donsker, 157 strong, 7, 45, 157, 172, 235 weak, 7, 157, 232 inverse relationship, 91, 220 between partial maxima and first passage times, 80, 91 between renewal and counting processes, 52 inverse renewal theory, 202 Jagers, 4, 51, 56 Janson, 6, 13, 30, 37, 38, 73, 77, 83, 87, 102, 106, 107, 112, 116, 119, 120, 122, 126, 130, 141, 148, 149, 168, 169, 178, 192, 219 Kac, 6, 7, 77 259 Kaijser, 129 Karamata, 237 Katz, 45, 190, 191, 195 Kemperman, 66 Kesten, 66 Kiefer, 141 Kingman, 88 Kolmogorov, 15–17, 29, 55, 89, 178, 220 ladder ascending, 5, 9, 69, 72 strong, 5, 69, 71–73, 82 weak, 71, 74 descending strong, 71 weak, 70 epoch, 5, 9, 69, 70, 83, 86, 104, 110, 137, 216 height, 70–72, 82, 89, 96, 101, 104, 106, 120, 137, 216 strong ascending, 70, 72, 82 variable, 2, 5, 29, 49, 69, 81, 82, 88, 89, 93, 96, 102 method, 82, 83, 99–101, 107, 163 Lai, 7, 18, 30, 35, 36, 63, 71, 90, 96, 98, 99, 106, 151, 152, 154, 175, 182, 203 Lalley, 66, 92, 107 Lamperti, 64 Larsson-Cohn, 182, 193, 200, 208, 214, 215 law of large numbers, 1, 9, 15, 18, 29, 32, 36, 45, 68, 75, 88, 111, 177, 220 convergence rate, 45, 190 converse, 29 moment convergence, 18, 39, 198, 200 law of the iterated logarithm, 1, 7, 9, 10, 44, 46, 65, 68, 108, 124, 155, 172, 178, 180, 184, 185, 207, 234, 235 Anscombe version of, 44 converse, 44 limit points, 44, 172 set of, 44, 173, 174, 235 Lindberger, 171, 174 Lindvall, 56, 233 Lipschitz continuous, 183 local limit theorem, 92, 112, 218 Lo`eve, 15, 224 260 Index Lorden, 105, 106 Lr -convergence theorem, 224 Maejima, 112 Marcinkiewicz, 22, 65, 81, 89, 90, 97, 178, 203, 209, 225, 227 Marcinkiewicz–Zygmund, 15, 20, 21, 32, 47, 117, 145, 150, 206, 209 Markov random walk, 219 Markov renewal theory, 66, 218 martingale, 47, 56, 225–228 convergence, 226 moment inequalities, 225 optional sampling theorem, 25, 28, 228 reversed, 47, 226 martingale proof of law of large numbers, 47 McDonald, 56 Meyer, 56 Miller, 130 Mohan, 57, 61, 64 moment convergence, 198, 200 moment generating function, 203 moments convergence, 4, 18, 190, 198, 200 finiteness, 83, 96, 142, 190, 191, 195, 196, 198 inequalities for martingales, 225, 227 for stopped random walks, 10, 28, 46 for sums of independent random variables, 225, 226, 236 Marcinkiewicz–Zygmund, 20–22, 225, 227 Mori, 112 Nagaev, 66 negative binomial process, 50, 52, 60, 62, 63, 111 Nerman, 105 Neveu, 26, 226 Ney, 56, 112 Niculescu, 57 nonarithmetic, 50, 51, 55–57, 59, 61–63, 67, 94, 95, 120, 216 nonlinear renewal theory, 142, 156, 175, 177 null hypothesis, 211, 212 number of renewals, 51, 79 expected, 53, 69 visits, 68, 69 Nummelin, 56 open-ended test, 204 optional sampling theorem, 25 Orey, 96 Ornstein, 4, 6, 69 overshoot, 103–105, 107, 111, 178, 185, 201, 216, 221 partial maxima, 5, 7, 49, 71, 72, 74, 80, 91, 115, 137 partial minima, 5, 71, 72, 74, 137 perturbed random walk, 175, 177 renewal process, 193 two-dimensional, 205, 206, 210, 212 Pluci´ nska, 130 point possible, recurrent, 4, 68 transient, 68 Poisson process, 50, 51, 56, 60, 62, 133, 134 Pollard, 55, 95 Port, Prabhu, 4, 51, 69, 72, 73, 77, 80, 95, 116, 133, 134 projections, 160, 166 Proschan, 130–132 Pyke, 21, 132 queueing theory, 1, 2, 134 random change of time, 158, 161, 170, 233 random index, 1, 3, 4, 9, 64 random walk, 1–7, 9, 10, 22, 30, 39, 49, 50, 65–74, 80, 176, 216 arithmetic, 50, 67, 69, 95, 110, 120 d-arithmetic, 50, 53, 55–57, 62, 94, 95, 101, 104 with span d, 50, 51 Bernoulli, 1, 50 symmetric, www.ebook3000.com Index classification, 66 coin-tossing, drifting, 5, to +∞, 68, 70, 110 to −∞, 68, 70, 71 maximum of, 71 nonarithmetic, 50, 67, 94–96, 101, 103, 120 oscillating, 5, 6, 72 randomly indexed, 1, 3, 4, recurrent, 6, 68 simple, 1, 66–69, 109, 113 symmetric, 1, 26, 70 stopped, 4, 10 transient, 5, 6, 67, 68 two-dimensional, 3, 6, 81, 115, 205, 209 with positive drift, 3, 5, 73, 77, 79, 115, 136, 137, 162, 169, 174 record times, 219, 220 record values, 219 records, 219 recurrent event, regeneration times, 219 regularly varying function, 61, 143, 179, 214, 216, 237, 238 relative compactness, 172–174, 235 reliability theory, 1, 2, 9, 131, 132 renewal counting process, 2, 3, 9, 10, 51, 52, 54, 57, 65, 72, 80, 82, 87, 88, 91, 137 Berry–Esseen theorem, 65 central limit theorem, 57 large deviation, 65 law of iterated logarithm, law of large numbers, 9, 57 moment generating function, 87 moments convergence, 59–60 finiteness, 52 uniform integrability, 58 renewal function, 5, 6, 52–54, 63, 64, 67, 94, 95 extended, 94 renewal measure, 67, 95 harmonic, 96 renewal process, 1, 4–6, 49–56, 65, 72, 79, 80, 89, 94, 101, 103, 115, 120, 125, 132, 133, 137, 160 261 age, 63 alternating, 3, 130 arithmetic, 50, 56 d-arithmetic, 53, 55, 56, 59, 61 with span d, 50 coupling proofs, 56 delayed, 65 integral equation, 53 lifetime, 64, 103 residual, 61, 63, 103 nonarithmetic, 50, 51, 55, 56, 59, 61–63 pure, 65 terminating, 66, 70 renewal theorem, 5, 9, 81, 93, 95 Blackwell’s, 64, 93, 96 random walk analogs, 93 elementary, 5, 64, 93, 96 random walk analog, 93 key, 57, 59 remainder term estimate, 95 renewal theory, 2, 5, 6, 9, 51, 54, 64–66, 76, 79, 83, 93, 112, 175 for oscillating random walks, for random walks on the real line, 65, 66, 80 with positive drift, 5, 79 in higher dimensions, 64 Markov, 66, 218 multidimensional, 66 multivariate, 66 nonlinear, 7, 142, 156, 176, 177 R´enyi, 3, 9, 16, 80, 117, 130, 216, 220 repeated significance test, 178, 203–205, 209 R´ev´esz, 217 Revuz, Richter, 4, 12, 13 Robbins, 25, 26, 45, 99, 101, 112 Root, 21 Ros´en, 71 Roulette, 134 Schwabe, 205, 211, 214 Seal, 133 Seneta, 143, 237 sequential analysis, 1, 2, 9, 10, 24, 26 sequential test, 204, 212 Serfozo, 65, 160, 172, 174 262 Index Siegmund, 7, 26, 101, 107, 112, 116, 120, 143, 144, 146, 148, 149, 156, 175, 179, 182, 185, 203, 205, 211 Skorohod, 161, 230–232, 235 slowly changing, 176 slowly varying function, 64, 123, 143, 179, 215, 237–239 representation formula, 239 Smith, 9, 51, 57, 66, 86, 87, 112, 116, 119, 120, 122 Snell, 56 Sparre Andersen, 4, 133 Spitzer, 4, 6, 66, 69, 71, 93, 96 square root boundary, 203, 204 Sreehari, 238 stable law, 6, 18, 61, 65, 92, 122, 139, 147, 178, 180, 184 domain of attraction of, 6, 61, 65, 92, 123, 180, 184 stable process, 161, 163, 167, 217 without negative jumps, 169 without positive jumps, 163, 168, 171 Stam, 66 Steinebach, 217, 218 Stone, 6, 95 stopped random walk, 4–6, 10, 19, 36, 45, 46, 49, 80, 81, 83, 96, 109, 116, 172, 178, 205, 225 central limit theorem, 16, 18, 225 complete convergence, 109 convergence rate, 109 law of large numbers, 18 law of the iterated logarithm, 10, 46 moments, 21, 22, 24, 26–31, 36, 39 convergence, 4, 39, 41, 81, 82, 96, 98, 225 perturbed random walk, 175, 177– 179, 182, 183, 186–188, 205, 206, 210, 212, 214 sum, 22 two-dimensional perturbed random walks, 205, 211 two-dimensional random walk, 115, 116, 167, 173 applications, 81 uniform integrability, 10, 19, 36, 96, 198–202 weak convergence, 168 stopping summand, 18, 42, 89, 97, 144 stopping time, 4, 5, 7, 10, 18, 19, 22, 26, 29, 30, 33, 36, 42, 45, 49, 53, 54, 64, 72, 74, 79, 81–83, 125, 135, 137, 141, 149, 169, 178, 206, 228, 236 difference, 214–216 storage and inventory theory, Stout, 7, 44, 234, 235 Strassen, 7, 44, 157, 172, 174, 234, 235 strong approximation, 173, 218, 235 strong invariance principles, 217 strong law, 15, 18, 19, 39, 47, 57, 65, 75, 80, 81, 88–91, 97–99, 111, 112, 115, 116, 138, 144, 153, 177, 178, 183, 186–190, 201, 206, 209, 210, 213, 215, 218, 220 Marcinkiewicz–Zygmund, 65, 117, 178, 179, 184, 186, 188, 189, 201, 209 sums of random variables, 9, 29, 42, 112 dependent, 112 independent, 112, 225, 226 m-dependent, 107, 112 non-i.i.d, 112, 155 stationary, 112 vector valued, 124 Szynal, 45 Tă acklind, 61, 102 Tak´ acs, 9, 57, 130 Teicher, 6, 7, 25, 147 Teugels, 61, 64 Thorisson, 56 tightness, 161, 229, 236 topology J1 -, 163–167, 230, 231, 233, 236 M1 -, 163, 165, 169, 171, 230, 231 graph, 230 parametric representation, 167 U -, 231, 236 Torr˚ ang, 45 two-dimensional perturbed random walk, 167, 173, 205, 206, 210–212 uniform continuity in probability, 16, 176 uniform integrability, 10, 30, 36, 41, 43, 60, 82, 96, 99, 147, 198, 223, 224 Vervaat, 162, 166, 173, 174, 232, 235 www.ebook3000.com Index von Bahr, 116, 133 Wainger, 66, 95, 112 Wald, 77 weak convergence, 168, 174, 229 in the J1 -topology, 165, 167, 230, 231, 236 in the M1 -topology, 165, 230, 231 in the U -topology, 231, 236 weak invariance principles, 217 Whitt, 161, 165–167, 174, 232, 233 Wiener, 217 measure, 158, 160, 230 263 process, 168, 169, 218 Wiener–Hopf factorization, 4, 49 Williamson, 56, 64, 112 Wintner, 44, 234, 235 Wolfowitz, 95, 141 Woodroofe, 7, 103–105, 107, 177, 182, 185, 191, 198, 203, 205, 211 Yahav, 66 Yu, 33, 35 Zygmund, 20–22, 65, 81, 89, 90, 97, 178, 203, 209, 225, 227 ... numbers and the De Moivre–Laplace central limit theorem for Bernoulli random walks and simple random walks, which provide information about the asymptotic behavior of such random walks Similarly, limit. .. interval and so on It turns out that the limit theorems mentioned above can be extended to random walks with random indices Frequently such limit theorems provide a limiting relation involving the randomly... which we turn our attention to randomly indexed random walks Now, in order to prove theorems on uniform integrability and moment convergence for randomly indexed random walks under minimal conditions
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