Description
Advanced Probability I Homework 3
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
h(x)dFn(x) →
Z
h(x)dF(x).
Q3: Let X1, X2, . . . be i.i.d. and have the standard normal distribution. It is known that
P(Xi > x) ∼
1
√
2πx
exp(−
x
2
2
) as x → ∞.where a(x) ∼ b(x) means a(x)/b(x) → 1 if x → ∞.
(i): Prove that for any real number θ,
P(Xi > x +
θ
x
)/P(Xi > x) → exp(−θ), as x → ∞
(ii) Show that if we define bn by P(Xi > bn) = 1/n,
P(bn( max
1≤i≤n
Xi − bn) ≤ x) → exp(−e
−x
).
(iii) Show that bn ∼ (2 log n)
1
2 and conclude max1≤i≤n Xi/(2 log n)
12 → 1 in probability.
Q4: Let X1, X2, …. be independent taking values 0 and 1 with probability 1/2 each. Let
X = 2P
j≥1 Xj/3
j
. Compute the characteristic function of X.
Q5: Let Sn = X1 + · · · Xn in the following problems.
(a): Suppose that Xi
’s are independent and P(Xi = i) = P(Xi = −i) = i−α
4
and
P(Xi = 0) = 1 −
i−α 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 P(Xi = 1) = 1
i = 1 − P(Xi = 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 X2
n + Y
2
n
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
1
n
1
X1
+ . . . +
1
Xn
.
1
Q8: Do some self-study and explain why the Stable distributions and Infinitely divisible
distributions bear such names.
2
