Q1: Let X,X1,X2,... be a sequence of random variables defined on the same probability space (Ω,F,P). Further, let g : R → R. Let Dg be the set of the discontinuity points of g. Assume that P(X ∈ Dg) = 0. Prove the following continuous mapping theorem for convergence in distribution: If Xn −D→ X, then g(Xn) −D→ g(X).
Q2: Suppose g,h : R → R are continuous with g(x) > 0, and |h(x)|/g(x) → 0 as |x| → 0. Let F,F1,F2,... be a sequence of distribution functions. Suppose Fn → F weakly and R g(x)dFn(x) ≤ C < ∞ uniformly in n. Prove
Z Z
h(x)dFn(x) → h(x)dF(x).
Q3: Let X1,X2,... be i.i.d. and have the standard normal distribution. It is known that
) as x → ∞.
where a(x) ∼ b(x) means a(x)/b(x) → 1 if x → ∞.
(i): Prove that for any real number θ,
, as x → ∞
(ii) Show that if we define bn by P(Xi > bn) = 1/n, P(bn(max Xi − bn) ≤ x) → exp(−e−x).
1≤i≤n
(iii) Show that and conclude max 1 in probability.
Q4: Let X1,X2,.... be independent taking values 0 and 1 with probability 1/2 each. Let X = 2Pj≥1 Xj/3j. Compute the characteristic function of X.
Q5: Let Sn = X1 + ···Xn in the following problems.
(a): Suppose that Xi’s are independent and and
for some nonnegative parameter α. Find an(α),bn(α) such that (Sn − an(α))/bn(α) ⇒ N(0,1) when α ∈ (0,1) and prove this CLT.
(b):Suppose that Xi’s are independent and = 0). Find an and bn such that (Sn − an)/bn ⇒ N(0,1) and prove this CLT.
Q6: Suppose that Xn and Yn are independent, and Xn → X∞ in distribution and Yn → Y∞ in distribution. Show that converges in distribution.
Q7: Let X1,X2,... be i.i.d. with a density that is symmetric about 0, and continuous and positive 0. Find the limiting distribution of
.
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Q8: Do some self-study and explain why the Stable distributions and Infinitely divisible distributions bear such names.
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