Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions (Encyclopedia of Mathematics and its Applications)

Main subject categories: • Asymptotic analysis • Heavy-tailed distributions • Random walks • Probability • Mathematical statisticsThis book focuses on the asymptotic behaviour of the probabilities of large deviations of the trajectories of random walks with 'heavy-tailed' (in particular, regularly varying, sub- and semiexponential) jump distributions. Large deviation probabilities are of great interest in numerous applied areas, typical examples being ruin probabilities in risk theory, error probabilities in mathematical statistics, and buffer-overflow probabilities in queueing theory. The classical large deviation theory, developed for distributions decaying exponentially fast (or even faster) at infinity, mostly uses analytical methods. If the fast decay condition fails, which is the case in many important applied problems, then direct probabilistic methods usually prove to be efficient. This monograph presents a unified and systematic exposition of the large deviation theory for heavy-tailed random walks. Most of the results presented in the book are appearing in a monograph for the first time. Many of them were obtained by the authors. Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Contents......Page 6 Notation......Page 12 Introduction......Page 16 1.1 Regularly varying functions and their main properties......Page 32 1.2 Subexponential distributions......Page 44 1.3 Locally subexponential distributions......Page 75 1.4 Asymptotic properties of ??functions of distributions?ˉ......Page 82 1.5 The convergence of distributions of sums of random variables with regularly varying tails to stable laws......Page 88 1.6 Functional limit theorems......Page 106 2.1 Introduction. The main approach to bounding from above the distribution tails of the maxima of sums of random variables......Page 111 2.2 Upper bounds for the distribution of the maximum of sums when |á  1 and the left tail is arbitrary......Page 115 2.3 Upper bounds for the distribution of the sum of random variables when the left tail dominates the right tail......Page 122 2.4 Upper bounds for the distribution of the maximum of sums when the left tail is substantially heavier than the right tail......Page 128 2.5 Lower bounds for the distributions of the sums. Finiteness criteria for the maximum of the sums......Page 134 2.6 The asymptotic behaviour of the probabilities P(Sn  x)......Page 141 2.7 The asymptotic behaviour of the probabilities P(Sn  x)......Page 151 3.1 Upper bounds for the distribution of Sn......Page 158 3.2 Upper bounds for the distribution of Sn(a), a > 0......Page 168 3.3 Lower bounds for the distribution of Sn......Page 172 3.4 Asymptotics of P(Sn  x) and its refinements......Page 173 3.5 Asymptotics of P(Sn  x) and its refinements......Page 180 3.6 The asymptotics of P(S(a)  x) with refinements and the general boundary problem......Page 185 3.7 Integro-local theorems on large deviations of Sn for index .|á, |á ?ê (0, 2)......Page 197 3.8 Uniform relative convergence to a stable law......Page 204 3.9 Analogues of the law of the iterated logarithm in the case of infinite variance......Page 207 4.1 Upper bounds for the distribution of Sn......Page 213 4.2 Upper bounds for the distribution of Sn(a), a > 0......Page 222 4.3 Lower bounds for the distributions of Sn and Sn(a)......Page 225 4.4 Asymptotics of P(Sn  x) and its refinements......Page 228 4.5 Asymptotics of P(Sn  x) and its refinements......Page 235 4.6 Asymptotics of P(S(a)  x) and its refinements. The general boundary problem......Page 239 4.7 Integro-local theorems for the sums Sn......Page 248 4.8 Extension of results on the asymptotics of P(Sn  x) and P(Sn  x) to wider classes of jump distributions......Page 255 4.9 The distribution of the trajectory {Sk} given that Sn  x or Sn  x......Page 259 5.1 Introduction......Page 264 5.2 Bounds for the distributions of Sn and Sn, and their consequences......Page 269 5.3 Bounds for the distribution of Sn(a)......Page 278 5.4 Large deviations of the sums Sn......Page 281 5.5 Large deviations of the maxima Sn......Page 299 5.6 Large deviations of Sn(a) when a > 0......Page 305 5.7 Large deviations of Sn(.a) when a > 0......Page 318 5.8 Integro-local and integral theorems on the whole real line......Page 321 5.9 Additivity (subexponentiality) zones for various distribution classes......Page 327 6.1 Introduction. The main method of studying large deviations when Cram??er?ˉs condition holds. Applicability bounds......Page 331 6.2 Integro-local theorems for sums Sn of r.v.?ˉs with distributions from the class ER when the function V (t) is of index from the interval (.1,.3)......Page 339 6.3 Integro-local theorems for the sums Sn when the Cram??er transform for the summands has a finite variance at the right boundary point......Page 346 6.4 The conditional distribution of the trajectory {Sk} given Sn ?ê |¤[x)......Page 349 6.5 Asymptotics of the probability of the crossing of a remote boundary by the random walk......Page 350 7.1 Functions of regularly varying distributions......Page 366 7.2 Functions of semiexponential distributions......Page 372 7.3 Functions of distributions interpreted as the distributions of stopped sums. Asymptotics for the maxima of stopped sums......Page 375 7.4 Sums stopped at an arbitrary Markov time......Page 378 7.5 An alternative approach to studying the asymptotics of P(Sn  x) for sub- and semiexponential distributions of the summands......Page 385 7.6 A Poissonian representation for the supremum S and the time when it was attained......Page 398 8.1 Introduction......Page 400 8.2 A fixed level x......Page 401 8.3 A growing level x......Page 422 9.1 Introduction......Page 429 9.2 Integro-local large deviation theorems for sums of independent random vectors with regularly varying distributions......Page 433 9.3 Integral theorems......Page 443 10.1 Introduction......Page 448 10.2 One-sided large deviations in trajectory space......Page 449 10.3 The general case......Page 453 11.1 The formulation of the problem for sums of random variables of two types......Page 458 11.2 Asymptotics of P(m, n, x) related to the class of regularly varying distributions......Page 460 11.3 Asymptotics of P(m, n, x) related to semiexponential distributions......Page 463 12.1 Upper and lower bounds for the distributions of Sn and Sn......Page 470 12.2 Asymptotics of the crossing of an arbitrary remote boundary......Page 485 12.3 Asymptotics of the probability of the crossing of an arbitrary remote boundary on an unbounded time interval. Bounds for the first crossing time......Page 488 12.4 Convergence in the triangular array scheme of random walks with non-identically distributed jumps to stable processes......Page 495 12.5 Transient phenomena......Page 502 13.1 Upper and lower bounds for the distributions of Sn and Sn......Page 513 13.2 Asymptotics of the probability of the crossing of an arbitrary remote boundary......Page 526 13.3 The invariance principle. Transient phenomena......Page 533 14.1 The classes of random walks with dependent jumps that admit asymptotic analysis......Page 537 14.2 Martingales on countable Markov chains. The main results of the asymptotic analysis when the jump variances can be infinite......Page 540 14.3 Martingales on countable Markov chains. The main results of the asymptotic analysis in the case of finite variances......Page 545 14.4 Arbitrary random walks on countable Markov chains......Page 547 15.1 Introduction......Page 553 15.2 The first approach, based on using the closeness of the trajectories of processes in discrete and continuous time......Page 556 15.3 The construction of a full analogue of the asymptotic analysis from Chapters 2 ̈C5 for random processes with independent increments......Page 563 16.1 Introduction......Page 574 16.2 Large deviation probabilities for S(T) and S(T)......Page 582 16.3 Asymptotic expansions......Page 605 16.4 The crossing of arbitrary boundaries......Page 616 16.5 The case of linear boundaries......Page 623 Bibliographic notes......Page 633 References......Page 642 Index......Page 655 This Monograph Is Devoted To Studying The Asymptotic Behaviour Of The Probabilities Of Large Deviations Of The Trajectories Of Random Walks, With 'heavy-tailed' (in Particular, Regularly Varying, Sub- And Semiexponential) Jump Distributions. It Presents A Unified And Systematic Exposition. Preliminaries -- Random Walks With Jumps Having No Finite First Moment -- Random Walks With Jumps Having Finite Mean And Infinite Variance -- Random Walks With Jumps Having Finite Variance -- Random Walks With Semiexponential Jump Distributions -- Large Deviations On The Boundary Of And Outside The Cramer Zone For Random Walks With Jump Distributions Decaying Exponentially Fast -- Asymptotic Properties Of Functions Of Regularly Varying And Semiexponential Distributions. Asymptotics Of The Distributions Of Stopped Sums And Their Maxima. An Alternative Approach To Studying The Asymptotics Of P(s[subscript N] [is Equal To Or Greater Than] X) -- On The Asymptotics Of The First Hitting Times -- Integro-local And Integral Large Deviation Theorems For Sums Of Random Vectors -- Large Deviations In Trajectory Space -- Large Deviations Of Sums Of Random Variables Of Two Types -- Random Walks With Non-identically Distributed Jumps In The Triangular Array Scheme In The Case Of Infinite Second Moment. Transient Phenomena -- Random Walks With Non-identically Distributed Jumps In The Triangular Array Scheme In The Case Of Finite Variances -- Random Walks With Dependent Jumps -- Extension Of The Results Of Chapters 2-5 To Continuous-time Random Processes With Independent Increments -- Extension Of The Results Of Chapters 3 And 4 To Generalized Renewal Processes. A.a. Borovkov, K.a. Borovkov ; Translated By O.b. Borovkova. Series Numbering From Jacket Includes Bibliographical References (p. 611-623) And Index.