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Many aspects of signal processing and communication theory have proven to be limited by a lack of sufficient developments in the areas of probability theory and mathematical statistics. Our investigations attempted to overcome this deficiency by contributing both to the underlying theoretical basis of the area as well as to some applied aspects of the area, and we have obtained a large body of results. Some principal thrust areas of our research effort have been concerned with quantization theory, signal detection, and estimation theory. A brief topical overview of our research areas follows. Quantization - existence of optimal quantizers, convergence properties of sequences of quantizers, design of quantizers, Signal detection - effects of statistical dependence, relative efficiency between detectors, nonparametric detection, Estimation - effects of imperfect data, a general class of fidelity criteria leading to nonlinear estimators, convergence properties of sequences of estimators; and Additional areas - meadian filters; stability of differential equations with random coefficients; bandwidth properties of random processes; contention resolution in local area computer networks; image compression; counterexamples.