An Invariance Principle for the Empirical Process with Random Sample Size

Let (B be the cr-field of Borel sets of C. Let (0, Ct, P ) be some probability space and W be the Wiener measure on (C, <£) with the corresponding Wiener process { W t ( o ) ) : 0 S t ^ l } , coGO; that is Wt has values in C and is specified by E(Wt)=0 and E(W.Wt)=s ifs^t. Let W° be the Gaussian measure on (C, (B) constructed by setting WÏ = Wt—tWi. Then W°tEC, E ( W ? ) = 0 and E(W^TF?) 5 ( 1 -* ) if s;g*. Also Wg = W? = 0 with probability 1 and {W^lO^t^l} is called the tied down Wiener process or the Brownian bridge. Let Sn = £i + • * • +£n, 5o = 0, » = 1, 2, • • • be the partial sum sequence of random variables {£M} defined on (Q, Cfc, P ) . Define a random element Xn of C by (1) Xn(t, co) = Wn(t, co) + (»< [nt])tM+i(f*)/nU* tWn(l, «) where PFnO, eo) = S[nt](co)/n . The following theorem is an immediate consequence of L. Breiman's analysis of §§13.5 and 13.6 in his book [3].