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.
The paper deals with the expected maxima of continuous Gaussian processes X = (X t ) t≥0 that are Hölder continuous in L 2 -norm and/or satisfy the opposite inequality for the L 2 -norms of their increments. Examples of such processes include the fractional Brownian motion and some of its "relatives" (of which several examples are given in the paper). We establish upper and lower bounds for E max 0≤t≤1 X t and investigate the rate of convergence to that quantity of its discrete approximation E max 0≤i≤n X i/n . Some further properties of these two maxima are established in the special case of the fractional Brownian motion.
For the fractional Brownian motion B H with the Hurst parameter value H in (0,1/2), we derive new upper and lower bounds for the difference between the expectations of the maximum of B H over [0,1] and the maximum of B H over the discrete set of values in −1 , i = 1, . . . , n. We use these results to improve our earlier upper bounds for the expectation of the maximum of B H over [0, 1] and derive new upper bounds for Pickands' constant.
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