The bootstrap for empirical processes under dependence.

In this thesis we establish that the blockwise bootstrap works for a large class of statistics. The main results are as follows: (i) A strongly mixing sequence satisfying the Central Limit Theorem for the mean, also satisfies the Moving Blocks Bootstrap Central Limit Theorem, in probability, even with bootstrapped norming. This result is optimal. (ii) The blockwise bootstrap of the empirical processes for a stationary sequence, indexed by VC-subgraph classes of functions, converges weakly to the appropriate Gaussian Process, conditionally in probability. The conditions imposed are only marginally stronger than the best known sufficient conditions for the regular CLT for these processes. (iii) The blockwise bootstrap of the classical empirical process for stationary strong mixing sequences, converges weakly to the appropriate Brownian bridge, conditionally in probability. The conditions imposed are weaker than the weakest known sufficient conditions for a regular CLT for this process.