A Toolbox for Refined Information-Theoretic Analyses with Applications

This monograph offers a toolbox of mathematical techniques that have been effective and widely applicable in information- theoretic analyses. The first tool is a generalization of the method of types to Gaussian settings, and then to general exponential families. The second tool is Laplace and saddle- point integration, which allow to refine the results of the method of types, and can obtain various precise asymptotic results. The third is the type class enumeration method, a principled method to evaluate the exact random-coding exponent of coded systems, which results in the best known exponent in various problems. The fourth is a subset of tools aimed at evaluating the expectation of non-linear functions of random variables, either via integral representations, by a refinement of Jensen's inequality via change-of-measure, by complementing Jensen's inequality with a reversed inequality, or by a class of generalized Jensen's inequalities that are applicable for functions beyond convex/concave. Various examples of all these tools are provided throughout the monograph.