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Example text
Define tj = j8, 1 :::; j :::; rand to = 0. Let f j = Wd(tj ) - Wd(tj-d , 1 :::; j :::; r. 7) ~ E/3) IWd(S) - Wd(tj - d l t j _ l ::;S::;t j = I n( 8) + II (8) , Gd{t)] + 2::= P[lfjl j= l ~ E/ 6] (say). lJ :::; Gd{l) j=l = 1. 2. 9) L aj :s; [Gd(t + 0) - Gd(t) ], all rand n . sup O::; t:9 - <5 j=l Furthermore, (Nl) and (N2) enable one to apply the Lindeberg Feller Central limit Theorem (L-F CLT) to conclude that atfj -+d Z , Z a N( o, 1) L V . 10) -+ 0 as n -+ 00 . 9) , lim sup n IIn (0) T < 3 lim sup L (6aj / E)4 (E Z 4 = 3) j=l n < IiE- 4 lim sup n sup [Gd(t + 0) - Gd(t )].
3 , respectively. 2) in u. 2. 6 to derive the limiting distribution of some tests for fitting an autoregressive model to the given time series . H. L. Koul, Weighted Empirical Processes in Dynamic Nonlinear Models © Springer-Verlag New York, Inc 2002 2. P. 2. This inequality is of a general interest . It is used to carry out a chaining argument pertaining to the weak convergence of Vh with bounded h. t . d . 's and a proof of its martingale property. 's of independent r. 's. This inequality is an extension of the well celebrated Dvoretzky, Kiefer and Wolfowitz (1956) inequality for the ordinary empirical process.
I nmrxd7[n-l L{Gi(t + <5)} i _n- 1 L {Gi (t ) - t} + <5] i < n max dT <5, I 0 ~ t ~ 1 - <5, by (D) . Thus (B) and (D) together imply (N2) and (C) . 1. 15) below) . This pro of will be bas ed on the following two lemmas. 13) Proof. Suppose 0 ~ s ~ t ~ 1. 1. 1. Weak convergence 23 OfWd +3 [ ~ d;Pi(1- Pi)] 2 z k~{ n- < But s :s: t 2 LPi +3 z (n-1~Pi) 2}. z and (D) imply a:s: n- 1 LPi = n- 1 L[Gi(t) - Gi(S)J i :s: (t - s). 13) :s: k~{n-l(t - s) + 3(t - sf} , o:s: s :s: t :s: 1. The proof is completed by interchanging the role of sand t in the above argument in the case t :s: s.