High resolution coding of stochastic processes and small ball probabilities

Abstract

In this thesis, we study the high resolution coding problem for stochastic processes. The general coding problem concerns the finding of a good representation of a random signal, the original, within a class of allowed representations. The class of allowed representations is defined through restrictions on the information content of these representations. Mostly we consider three different interpretations of information. In the quantization problem we constrain the reconstruction corresponding to a representation, which itself is a random element, to be supported by a finite set having less than er elements. The entropy coding problem restricts the entropy of the reconstruction to be less than r. Finally, the algorithmic coding problem constrains the Shannon mutual information between the original and the reconstruction to be less than r. This thesis is concerned with the asymptotic quality of the optimal coding scheme when the bound on the allowed information r tends to infinity: the high resolution coding problem. Our analysis considers Gaussian processes for the original and norm-based distortion measures. This means the distortion between the reconstruction and the original is measured as a power of the distance. A typical example is Wiener measure on the Banach space of continuous functions with corresponding norm. We derive asymptotic bounds for the high resolution coding problems. Our bounds are weakly and strongly tight for a broad class of originals in Banach and Hilbert spaces, respectively. Moreover, in the typical Hilbert space setting we show that the above three coding problems yield the same asymptotics. A further result concerns the efficiency of quantization with randomly generated codebooks instead of deterministic codebooks. It is found that under certain regularity conditions, which are fulfilled in the typical Hilbert space setting, the corresponding asymptotics are the same. A further objective is the effect of perturbations on the coding problem. These results yield a relation between the coding complexities of diffusions and Brownian motion in the entropy and algorithmic sense. A second subject of this thesis is the study of small ball probabilities around random centers. We find basic properties and estimates. Moreover, we obtain that in the Hilbert space setting the random small ball probabilities are asymptotically equivalent to a deterministic function. A similar statement is proven to hold for Wiener measure in the Banach space of continuous functions. Finally, the asymptotics of random small ball functions are related to a particular coding problem.