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STATISTICS 381: MEASURE-THEORETIC PROBABILITY I
REVIEW EXERCISES
Problem 1. True or False:
(A) If µn and µ are Borel probability measures on R such that µn =⇒ µ as n → ∞, then
Z
Z
lim
n→∞
x d µn (x ) =
x d µ(x ).
(B) If X 1 , X 2 , X 3 , . . . are independent, identically distributed random variables with the standard Gaussian distribution then for every α > 0 and every t ∈ R,
lim
n →∞
n
X
k −α X k sin(k t )
k =1
exists and is finite with probability one.
(C) If µn and µ are Borel probability measures on R such that µn =⇒ µ as n → ∞, then
µ̂n (θ ) → µ̂(θ ) uniformly for all θ ∈ R.
Problem 2. Let X 1 , X 2 , X 3 , . . . be independent, identically distributed random variables
with common distribution
1
1
and P {X i = −m } =
for m = 0, 1, 2, . . . .
P {X i = +1} =
m
2
2 +1
Define
n
X
Xi ,
Sn =
i =1
T+ = min{n ≥ 1 : Sn = 1},
T− = min{n ≥ 1 : Sn ≤ −1}.
(A) Evaluate E T+ .
(B) Evaluate E s T+ for 0 < s < 1.
(C) Evaluate E s T− for 0 < s < 1.
Problem 3. Let X 1 , X 2 , X 3 , . . . be independent, identically distributed Rademacher− 21 random variables , and let a 1 , a 2 , a 3 , . . . be a bounded sequence of real numbers such that
n
1X 2
lim
a i = σ2 .
n→∞ n
i =1
1
(A) Prove that as n → ∞
n
1X
a i X i −→ 0
n i =1
(B) Prove that as n → ∞
almost surely.
n
1 X
a i X i =⇒ Normal(0, σ2 ).
p
n i =1