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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.

Considered is quantum tunnelling in anisotropic spin systems in a magnetic field perpendicular to the anisotropy axis. In the domain of small field the problem of calculating tunnelling splitting of energy levels is reduced to constructing the perturbation series with degeneracy, the order of degeneracy being proportional to a spin value. Partial summation of this series taking into account "dangereous terms" with small denominators is performed and the value of tunnelling splitting is calcaulated with allowance for the first correction with respect to a magnetic field.

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In the three-dimensional holography of phase objects, scatter screens have been used to provide m u s e illumination. In the reconstruction process a diaphragm is used to select a particular angle or position of view. The paper shows that there are upper and lower limits to the size of the diaphragm, determined by the optical path variations within the object. These limits are not necessarily compatible with'one another. Thus the scatter screen technique is shown to have limitations. An alternative is simply to use an array of parallel beams separated by discrete angular intervals. The advantage of continuous variation of viewing angle is lost but the resistrictions on the type of object are less severe. In addition the reconstructions may take the form of schlieren or shadow photographs as well as interferograms.

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