Math/Stat 831 – Fall 2008
Final Exam
1. (9 points) Let X1, X2, . . . be i.i.d. with mean 0 and variance 1. Let anbe an increasing sequence of
positive numbers, and denote An= (Pn
k=1 a2
n)1/2. Show that if an/An→0 then
n
P
k=1
akXk
An
⇒N(0,1),
i.e. the normalized sums converge to a standard normal random variable in distribution.
2. (10 points) Suppose that for two random variables Xand Ythe following statements hold:
•X−Yand Yare independent,
•X−Yand Xare independent.
(a) Show that if the characteristic functions of Xand X−Yare denoted by ψX(t) and ψX−Y(t) then
ψX(t) (1 − |ψX−Y(t)|2) = 0.(1)
(b) Show that if equation (1) holds for all tthen |ψX−Y(t)|= 1 if tis close enough to 0.
(c) Show that X−Yis constant with probability 1.
3. (10 points) Let ξnbe independent random variables with distribution
ξn=
1 with probability (2n)−1
0 with probability 1 −1/n
−1 with probability (2n)−1
Let X0= 0 and define Xnrecursively by
Xn+1 =|ξn|nXn+ξn1(Xn= 0).
(a) Show that Xnis a martingale.
(b) Show that Xn
P
−→ 0. (Hint: it’s easy to estimate P(Xn= 0). . . )
(c) Show that Xndoes not converge to 0 a.s. (Hint: then ξnshould be 0 eventually.)
4. (9 points) Consider the Lebesgue probability space on the interval [0,1). (I.e. the state space is
Ω = [0,1), the σ-field is the set of Lebesgue measurable sets and the measure is the Lebesgue measure.)
We define the random variable Xas
X(ω) = (2ω, if 0 ≤ω < 1/2,
2ω−1 if 1/2≤ω < 1.
Compute the conditional expectation E(Y|X) where Y: [0,1) →Ris a measurable function.
5. (12 points) Let X1, X2, . . . be i.i.d. with P(Xn= 1) = p, P(Xn=−1) = 1 −pwhere 1/2< p < 1 is
a fixed constant. Let Sn=Pn
k=1 Xn(with S0= 0) and ϕ(x) = (1/p −1)x.
(a) Show that ϕ(Sn) is a martingale with respect to the natural filtration.
(b) For given a, b positive integers consider the first time Snreaches aor −b:
τ= inf{k:Sk=aor Sk=−b}.
Show that τis a stopping time with respect to the natural filtration.
(c) Show that Eϕ(Sτ) = 1 and compute the probability of reaching abefore b, i.e. P(Sτ=a).
(d) Find a suitable martingale to compute Eτ. (Hint: what is the expectation of Xn?)
Please present your solutions in a clear manner. Show all your work.