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Now, associated with 0 there is a number N such that n > N implies ISn - sol < 0, and hence o< To show that the "if" part is true, we first see that for all sequences converging to So> the corresponding Cauchy sequences have the same limit. Suppose that {Sn}neN and {tn}neN satisfy the above conditions and the sequences {X(Sn)}neN and {X(tn)}neN converge in L2(Q) to, say, Xo and Y, respectively. We can show Xo = Y by means of the following argument. Since the sequence Sb t1> S2, t2, ... ,Sn, tn, ...

S. continuous at t E [if and only if E {X (t1) X (t2)} is continuous at (t, t). s. continuous on [if and only if E {X(t1)X(t2)} is continuous on [2. Proof. The "only if' parts of (a) and (b) follow from the continuity of the inner product property discussed in Sect. 1. The "if' parts are true since E{X(S)_X(t)}2=EX2(S)-2E{X(s)X(t)}+EX2(t)~0 as s~t. 0 As shown in Sect. s. continuous second-order process is the Wiener-Levy process W: [0, T] ~ L2 (D) since, for each t E [0, T], IIW(s) - W(t)11 ~ ° as s ~ tin [0, T] .

Thus, if n is sufficiently large, Ll(Pn) < 0, and finally ti L If(tf) I 4-1f IIX'(t) i=1 = M b f edt = a - X'(tf) II dt ~ M k ti L f edt i=1 4-1 Me(b - a) o where M = max If(t)l. This completes the proof. 4 A Weak Condition for Existence Before proceeding, some extensions of the notion of partition disussed at the beginning of Sect. 4 are in order. Consider intervals [a, b] and [c, d] of JR and let R = [a, b] X [c, d] be a rectangle of JR z. If Ps = {[Si-I> sa, i = 1, .. ,m} u {sf, i = 1, .. ,m} is a partition of [a, b] and Pt = {[tj-I> tj],j = 1, ..

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