Probability Prelim, August 2010
1. Let Xi,i= 1,2, ..., be independent identically distributed (i.i.d.) random variables. Show that
a) if E|X1|<∞then
lim
n→∞ max
1≤i≤n
|Xi|
n= 0,a.s.and in L1
and
b) if moreover EX1= 0, then
lim
n→∞ max
1≤j≤n
|Sj|
n= 0 a.s.and in L1,
where Sj=Pj
i=1 Xi.
Hint: Whereas the first result can b e proved from first principles, the second requires use of a basic theorem
in Probability Theory. Recall that uniform integrability helps with convergence of moments.
2. a) State the three series theorem.
b) If Uiare i.i.d. random variables uniformly distributed on [−1,1], for what values of α > 0 does the series
PnUn/nαconverge a.s.? Same question for PnVn/nαif the variables Vnare uniformly distributed on [0,1].
3. a) Prove the central limit theorem for Xii.i.d. with EXi= 0, EX 2
i<∞. (Any proof that does not use
another CLT is acceptable.) State carefully any theorems you use in your proof.
b) If Ynis Poisson with expected value n, then find values of anand bnso that the random variables
bn(Yn−an) converge in distribution to the standard normal law.
4. Let (Ω,F,P) be the unit interval, with sigma algebra the Borel sets and with probability measure
equal to Lebesgue measure. For each n, let Fnbe the sigma algebra generated by the partition Ak,n =
[k/2n,(k+ 1)/2n), k= 0,...,2n−1, of [0,1), and let fbe a Lipschitz function on [0,1), that is, a function
such that |f(t)−f(s)| ≤ c|t−s|for a fixed c < ∞and all s, t ∈[0,1). For each n∈N, let the random
variables Xnon (Ω,F,P) be defined as
Xn(ω) =
2n−1
X
k=0
2n(f((k+ 1)/2n)−f(k/2n)) IAk,n (ω), ω ∈[0,1),
(that is, Xnis constant on each interval Ak,n, with value equal to the incremental quotient of fbetween the
end points of the interval).
a) Prove that (Xn,Fn), n∈N, is a martingale sequence.
b) Prove that there exists a random variable X∞such that Xn→X∞almost surely and in L1, with X∞
bounded by the Lipschitz constant c, and state the martingale convergence theorem you are using.
c) Prove that for all 0 ≤a≤b≤1,
f(b)−f(a) = Zb
a
X∞(t)dt
(hint: do this first for diadic numbers a=k/2nand b= (k+`)/2n, and then approximate).
