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Category: MATH 5411

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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)

1

2 → 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

2

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

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