Some results on empirical processes and stochastic complexity

This thesis contains two parts. In Part I, an intuitive and simple approach is used to obtain uniform convergence and Central Limit Theorems for empirical processes indexed by families of functions, for stationary mixing sequences with not too slow a $\beta$-mixing (or $\phi$-mixing) rate. The conditions imposed are in terms of random metric entropies, and the $\beta$-mixing (or $\phi$-mixing) rate. In the case of a Vapnik-Cervonenkis class, the uniform convergence result holds for any strictly stationary sequence with $\beta$-mixing (or $\phi$-mixing) rate $\beta\sb{n}$ (or $\phi\sb{n}$) = $o$($n\sp{-d}$) for some d $>$ 0, while the Central Limit Theorem requires the rate $\beta\sb{n}$ (or $\phi\sb{n}$ = $o$($n\sp{-d}$) for some d $>$ 3. In Part II, we use a shortest code length criterion to select the bin width of a histo-gram from a string of i.i.d. observations. Under smoothness conditions on the underlying density on (0, 1), this criterion is shown to be optimal as a code selection criterion in reaching the smallest redundancy within the class of codes under consideration. Furthermore, the criterion consistently selects the underlying density when it is uniform on (0, 1). Minimax lower bounds for the redundancy over many smooth families of underlying densities are also derived. In a special case, the lower bound is shown to be achieved almost surely, and in expectation, by a predictive code based on a sequence of histograms. In the case of Sobolev balls, the minimax optimal code is described and a heuristic proof is provided.