Math/Physic/Economic/Statistic Problems
1. (Not to turn in) Prove the Second Borel-Cantelli lemma: Suppose that An are independent
events, then if ∑P(An) = ∞ we have P(An i.o.) = 1 [Hint: 1 −x ≤e−x ∀x by convexity of
e−x.]
2. Prove that if (Xn) is a sequence of random variables with
limn→∞m→∞
P( maxm<k≤n|Xk −Xm|≥t) = 0 for all t > 0 then there is an X such that Xn →X (Varadhan Exer 3.12)
3. (not to turn in) Show that the Strong law holds for iid random variables assuming a finite
fourth moment bound.
Problem 2. Let (Xn)n∈Z be iid {±1}-valued Ber(1/2) random variables, i.e., P(X = 1) =
1/2,P(X = −1) = 1/2. Let
`n = max{m : Xn−m+1 = … = Xn = 1}
be the length of the run of +1’s at time n (if Xn = −1, this is zero).
1. Prove that
limn→∞
`n
log2 n = 1
limn→∞
`n = 0.
2. (not to be turned in) What does the first say about winning streaks?
Problem 3. Suppose that Xi are i.i.d. with EXi = 0 and V ar(Xi) = C < ∞ and Sn = ∑n i=1 Xi . Show that if we take n = mα where α > 1
2p−1 then
Sn
np →0
for p > 1/2 as m →∞. [Hint: Recall Kronecker’s lemma which says that if an →∞ and ∑
n≥1 xn
an
converges then
∑n
i=1 xi
n →0]
1
Problem 4. Let Xi be iid and Sn = ∑n
i=1 Xi. Show that if Sn
n →0 in probability then
max1≤m≤n Sm
n →0
in probability. [Hint: Use Levy’s maximal inequality]
Problem 5. Show the following
1. Use Fubini’s theorem to relate EY to ∫ P(Y ≥t)dt for a non-negative random variable Y .
2. Use this to show that if (Yi) are iid copies of a non-negative random variable Y then
Yi
n →
{
0 if EY < ∞
∞ if EY = ∞
almost surely.
3. Let Sn = ∑n
i=1 Xi, where Xi are iid random variables. If Sn
n1/p →0 a.s., then E|X1|p < ∞ for
