Central Limit Theorems for Empirical Processes Based on Stochastic Processes

In this thesis, we study time-dependent empirical processes, which extend the classical empirical processes to have a time parameter; for example the empirical process for a sequence of independent stochastic processes {Yi : i ? N}: (1) ?n(t, y) = n^(?1/2 )Sigma[1(Y i(t)?<=y) ? P(Yi(t) <= y)] from i=1 to n, t ? E, y ? R.

In the case of independent identically distributed samples (that is {Yi(t) : i ? N} are iid), Kuelbs et al. (2013) proved a Central Limit Theorem for ?_n(t, y) for a large class of stochastic processes.

In Chapter 3, we give a sufficient condition for the weak convergence of the weighted empirical process for iid samples from a uniform process: (2) ?n(t, y) := n^(?1/2 )Sigma[w(y)(1(X (t)<=y) ? y)] from i=1 to n, t ? E, y ? [0, 1] where {X (t), X1(t), X2(t), ? ? ? } are independent and identically distributed uniform processes (for each t ? E, X (t) is uniform on (0, 1)) and w(x) is a ?weight? function satisfying some regularity properties. Then we give an example when X (t) := Ft(Bt) : t ? E = [1, 2], where Bt is a Brownian motion and Ft is the distribution function of Bt.

In Chapter 4, we investigate the weak convergence of the empirical processes for non-iid samples. We consider the weak convergence of the empirical process: (3) ?n(t, y) := n^(?1/2 )Sigma[(1(Y (t)<=y) ? Fi(t,y))] from i=1 to n, t ? E ? R, y ? R where {Yi(t) : i ? N} are independent processes and Fi(t, y) is the distribution function of Yi(t). We also prove that the covariance function of the empirical process for non-iid samples indexed by a uniformly bounded class of functions necessarily uniformly converges to the covariance function of the limiting Gaussian process for a CLT.