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The ξn are iid N(0, 1) random variables. 5) is called the KKL expansion for Xt . KKL Expansion for Brownian Motion Since R(t, s) = min {t, s}, 0 ≤ t, s ≤ 1, we first find the eigenfunctions and eigenvalues of R. To do this, consider the integral equation 1 min {t, s} φ(s)ds = λφ(t). 6) 0 That is, t λφ(t) = 1 sφ(s) ds + t φ(s) ds. 7) t 0 The right side shows that φ is differentiable and 1 λφ (t) = tφ(t) + 1 φ(s)ds – tφ(t) = t φ(s) ds. 8) t From this, we obtain λφ (t) = –φ(t). For convenience, write μ = λ1 .

Define the process Yn (t, ω) = j∈In Xnj (ω)Gnj (t). Note that the effect of each Xnj is 1 localized to a time-interval of length 2n–1 . Set Mn = maxj∈In |Xnj | and Ln = maxj∈In Xnj . For any positive number a, and n ≥ 1, we have P {Mn > a} ≤ 2P {Ln > a} by symmetry of Xnj = 2P e > e Ln a ≤ 2 Ln Ee ea ≤ 2 n–1 1/2 2 e ea ≤ since eLn ≤ eXnj j∈In 2n+1 . ea Choosing a = 2(n + 1) log 2, we get P {Mn > a} ≤ 2–(n+1) and hence, ∞ P {Mn > a} < ∞. n=1 By the first part of the Borel-Cantelli lemma, P Mn ≤ 2(n + 1) log 2 for all large enough n = 1.

Ii) For any Borel set B ∈ B(Rn ), assign P (X1 , . . , Xn ) ∈ B = 1 if (1, . . , 1) ∈ B and equal to zero otherwise. Prove that P is additive on A but there is no extension of P to a probability measure on F. 18 | Stochastic Processes 2. Let P and Q be two probability measures on ( , F ). Let S be a π system such that F = σ (S). If P = Q on S, show that P = Q on F . 3. Let S be as in the previous problem. Suppose L is a space of F -measurable functions such that (i) 1 ∈ L; IA ∈ L ∀A ∈ S, (ii) f , g ∈ L, then af + bg ∈ L for all nonnegative constants a, b, and (iii) If fn is a non-decreasing sequence of nonnegative functions in L such that limn→∞ fn = f , then f ∈ L.