Asymptotic Analysis of Random Walks: Light-Tailed Distributions (Encyclopedia of Mathematics and its Applications, Series Number 176)

Main subject categories: • Asymptotic analysis • Light-tailed distributions • Random walks • Asymptotic expansions • Asymptotic distribution ‒ Probability theory • Probability • Mathematical statisticsThis is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time. Copyright Contents Introduction 1 Preliminary results 1.1 Deviation function and its properties in the one-dimensional case 1.2 Deviation function and its properties in the multidimensional case 1.3 Chebyshev-type exponential inequalities for sums of random vectors 1.4 Properties of the random variable γ = Λ(ξ) and its deviation function 1.5 The integro-local theorems of Stone and Shepp and Gnedenko’s local theorem 2 Approximation of distributions of sums of random variables 2.1 The Cramér transform.The reduction formula 2.2 Limit theorems for sums of random variables in the Cramér deviation zone. The asymptotic density 2.3 Supplement tosection 2.2 2.4 Integro-local theorems on the boundary ofthe Cramér zone 2.5 Integro-local theorems outside the Cramér zone 2.6 Supplement to section 2.5. The multidimensional case. The class of distributions ER 2.7 Large deviation principles 2.8 Limit theorems forsums of random variables with non-homogeneous terms 2.9 Asymptotics of the renewal function and related problems. The second deviation function 2.10 Sums of non-identically distributed random variables in the triangular array scheme 3 Boundary crossing problems for random walks 3.1 Limit theorems for the distribution of jumps when the end of a trajectory is fixed. A probabilistic interpretation of the Cramér transform 3.2 The conditional invariance principle and the law of the iterated logarithm 3.3 The boundary crossingproblem 3.4 The first passage time of a trajectory over a high level and the magnitude of overshoot 3.5 Asymptotics of the distribution of the first passage time through a fixed horizontal boundary 3.6 Asymptotically linear boundaries 3.7 Crossing of a curvilinear boundary by a normalised trajectory of a random walk 3.8 Supplement. Boundary crossing problems in the multidimensional case 3.9 Supplement. Analytic methods for boundary crossing problems with linear boundaries 3.10 Finding thenumerical values of large deviation probabilities 4 Large deviation principles for random walk trajectories 4.1 On large deviation principles in metric spaces 4.2 Deviation functional (or integral) for random walk trajectories and its properties 4.3 Chebyshev-type exponential inequalities for trajectories of random walks 4.4 Large deviation principles for continuous random walk trajectories. Strong versions 4.5 An extended problem setup 4.6 Large deviation principles in the space of functions without discontinuities of the second kind 4.7 Supplement. Large deviation principles in the space (V,ρV) 4.8 Conditional large deviation principles in the space(D,ρ) 4.9 Extension of results to processes with independent increments 4.10 On large deviation principles for compound renewal processes 4.11 On large deviation principles for sums of random variables defined on a finite Markov chain 5 Moderately large deviation principles for the trajectories of random walks and processes with independent increments 5.1 Moderately large deviation principles for sums Sn 5.2 Moderately large deviation principles for trajectories sn 5.3 Moderately large deviation principles for processes with independent increments 5.4 Moderately large deviation principle as an extension of the invariance principle to the large deviation zone 5.5 Conditional moderately large deviation principles for the trajectories of random walks 6 Some applications to problems in mathematical statistics 6.1 Tests for two simple hypotheses. Parameters of the mostpowerful tests 6.2 Sequential analysis 6.3 Asymptotically optimal non-parametric goodness of fit tests 6.4 Appendix. On testing two composite parametric hypotheses 6.5 Appendix. The change point problem Basic notation References Index "This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time"-- Provided by publisher This is a complete and systematic modern treatise on large deviation theory for random walks with light-tailed jump distributions, presented by one of its key creators. Such distributions have numerous applications in statistics, ruin theory, and queuing theory. This is a companion to the author's earlier monograph on heavy-tailed distributions. 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.