09HW2solutions

# 09HW2solutions - HOMEWORK 2 SOLUTIONS Problem 1 Fix i1 i2...

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HOMEWORK 2 SOLUTIONS Problem 1 Fix i 1 ,i 2 ,...,i k ∈ I . We shall show that Y i 1 ,...Y i k are independent by showing that P { Y i 1 t i ,...,Y i k t k } = k Y l =1 P ( Y i l t l ) , t 1 ,t 2 ,..t k R . Step1: For any bounded continuous functions f 1 ,f 2 ,...,f k , we have E k Y l =1 f l ( Y i l ) ! = k Y l =1 E f l ( Y i l ) . X n,i l a.s. Y i l implies that f l ( X n,i l ) a.s. f l Y i l , and since all the functions are bounded, Bounded Convergence Theorem (or DCT) can be applied: E k Y l =1 f l ( Y i l ) ! = lim n + E k Y l =1 f l ( X n,i l ) ! = lim n + k Y l =1 E f l ( X n,i l ) ! = k Y l =1 E f l ( Y i l ) , where the equality before last comes from the independence of { X n,i l } k l =1 . Step 2: Consider the function f l,m = 1 , if - ∞ ≤ x t l - m ( x - t l ) + 1 , if t l x t l + m - 1 0 , if t l + m - 1 x < + Then f l,m ( x ) m →∞ -→ 1 ( -∞ ,t l ] ( x ) . All functions are bounded by 1

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09HW2solutions - HOMEWORK 2 SOLUTIONS Problem 1 Fix i1 i2...

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