Asymptotic Expansions for Pseudodifferential Operators on by Harold Widom

By Harold Widom

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Lim supn→∞ |X ln n Let X be a nonnegative random variable. (i) Show that n P(X > n) → 0 as n → ∞ if EX < ∞. ] (ii) Prove that ∞ n=1 P(X > ∞ n) ≤ EX ≤ ∞ n=0 P(X > n). [Hint: n=1 (n − 1)P(n − 1 < X ≤ n) ≤ EX ≤ ∞ n=1 n P(n − 1 < X ≤ n). ] Let Un = (Un,1 , . . , Un,n ), be uniformly distributed over the n-dimensional cube Cn = [0, 2]n for each n = 1, 2, . . That is, the distribution of Un is 2−n 1Cn (x)m n (dx), where m n is n-dimensional Lebesgue measure. Define X n = Un,1 · · · Un,n , n ≥ 1. Show that (a) X n → 0 in probability as n → ∞, [Hint: Compute EX nt as an iterated integral for strategic choices of t > 0], and (b) {X n : n ≥ 1} is not uniformly integrable; Suppose U is uniformly distributed on the unit interval [0, 1].

10(b) yields the so-called disintegration formula E( f (Y )) = Ω f (y)Q G (ω, dy)P(dω). 10 follows from the existence of a regular conditional distribution of X , given G. The following simple examples tie up the classical concepts of conditional probability with the more modern general framework presented above. Example 6 Let B ∈ F be such that P(B) > 0, P(B c ) > 0. Let G = σ(B) ≡ {Ω, B, B c , ∅}. Then for every A ∈ F one has 5 The Doob–Blackwell theorem provides the existence of a regular conditional distribution of a random map Y , given a σ-field G , taking values in a Polish space equipped with its Borel σ-field B(S).

As n → ∞. 6. Suppose that X n , n ≥ 1, is a sequence of random variables that converge to X in probability as n → ∞, and g is a continuous function. Show that g(X n ), n ≥ 1, converges in probability to g(X ). 7. Suppse that X n , n ≥ 1 and Yn , n ≥ 1, converge in probability to X and Y , and X n − Yn → 0 in probability as n → ∞, respectively. s. 8. Suppose that X n , n ≥ 1, is a sequence of real-valued random variables such that |X n | ≤ Y on Ω with EY < ∞. Show that if X n → X in probability as n → ∞, then EX n → EX as n → ∞.