December 2009,
Preliminary Exam: Probability, Friday, January 8, 2010
The exam lasts from 9:00 am until 2:00 pm. Your goal on this exam should be
to demonstrate mastery of probability theory and maturity of thought. Your arguments
should be clear, careful and complete. The exam consists of 6 problems. The several
steps that the problems are made of are designed to help you in the overall solution. If
you cannot justify a certain step, you still may use it in a later step.
On each page you turn in, write your assigned code number instead of your name.
Separate and staple each problem and return it to its designated folder.
Problem 1. Let X and Y be two positive and independent random variables.
∞
2
(a) (5 points) Prove E[(min{ X , Y }) ] =
∫ 2 zP( X
> z ) P (Y > z )dz
z =0
(b) (5 points) Prove or disprove: If E ( X ) = ∞ and E (Y ) = ∞ then
E[(min{ X , Y }) 2 ] = ∞
2
Problem 2. Let X 1 , X 2 ,... be i.i.d. random variables with E ( X i ) = 0 and E ( X i ) = 1 .
Denote S n = X 1 + ... + X n . Assume that for any t > 0 , N t is a positive integervalued random variable. Prove the following statements:
(a) (8 points) If for some positive integer-valued function a (t ) which satisfies
lim a (t ) = ∞ and
t →∞
S N t − S a (t )
a (t )
Nt
→ 1 in probability then as t → ∞
a (t )
→ 0 in probability.
[Hint: You may want to consider applying an appropriate maximal inequality on the
event: a (t )(1 − δ ) ≤ N t ≤ a (t )(1 + δ ) .]
(b) (6 points) Show by using (a) that
SNt
a (t )
⇒ N (0,1) as t → ∞ .
(c) (6 points) Let Y1 , Y2 ,... be i.i.d. positive random variables with E (Yi ) = µ and
Var (Yi ) = 1 . Let Tn = Y1 + ... + Yn and N t = sup{k : Tk ≤ t} . Apply (b) to prove
µ ⋅ N t − TN t
⇒ N (0,1) as t → ∞ .
t/µ
N
1
[Hint: You may use the LLN for N t without proof. It says that t →
almost
t
µ
surely as t → ∞ .]
(d) (6 points) We continue with the setup of part (c).
Y
max { m } → 0 in probability as t → ∞ .
1≤ m ≤ 2t / µ
t
TN t − t
(ii) Show how to use (i) in order to get
→ 0 in probability as t → ∞ .
t
[ Hint: 0 ≤ t − TN ≤ YN +1 ].
t
t
(i) Prove that
(iii) Prove
µ ⋅ Nt − t
⇒ N (0,1) as t → ∞ .
t/µ
Problem 3. (15 points) Let Fk , k = 0,..., n be an increasing sequence of
σ − algebras.
Let H k ∈ Fk , k = 0,..., n − 1, ξ ∈ Fn be integrable random variables, and H n ≡ 0 .
Find Yk ∈ Fk , Z k ∈ Fk , k = 0,..., n (in terms of H k , k = 0,..., n and ξ )
so that the following holds
n
(a) {Z k } is a martingale with respect to {Fk }k = 0 .
(b) Y0 = Z 0 and Yn = ξ .
(c) Yk +1 − Yk = H k + Z k +1 − Z k , k = 0,..., n − 1
Problem 4. Let { X k , Fk }, k = 0, 1,... be a SUPERMG with X 0 =0 and
X k +1 ≥ X k − 1 . Let b < 0 < a and let τ = min{0 ≤ k : X k > a or X k < b}
(τ = ∞ if no such k exists).
(a) (6 points) Prove X τ = lim X τ ∧ n exists a.s. ( Hint: X τ ∧n is bounded from
n→∞
below. Warning: we may have P(τ = ∞ ) > 0 .)
(b) (5 points) Prove that lim inf E ( X τ ∧ n ) ≥ E ( X τ ) . Can you conclude that
n→∞
E( Xτ ) ≤ 0 ?
(c) (6 points) Prove: P( X τ ≥ a ) ≤
1− b
a +1− b
Problem 5. Let X n ⇒ X where the symbol ⇒ means convergence in distribution. In
D
what follows the symbol = means equal in distribution.
D k
(a) (6 points) Let k be a positive integer. Assume that for each n X n = ∑ X n , i ,
i =1
where { X n ,1 ,..., X n,k } are i.i.d. Prove that { X n ,1 , n = 1,2,...} is tight.
(b) (5 points) We continue with the setup of part a. Prove that the following
D k
representation holds: X = ∑ X i with { X 1 ,..., X k } i.i.d.
i =1
D n
(c) (6 points) Prove that if for each n X n = ∑ X n , i , where { X n ,1 ,..., X n ,n } are
i =1
D k
i.i.d. , then for each k we have the representation X = ∑ Yk , i with
i =1
{Yk ,1 ,..., Yk , k } i.i.d.
Problem 6. In what follows
function of the r.v. X.
(a) (5 points) Prove: If
ϕ X (t ) = E (e itX ), -∞ < t < ∞ represent characteristic
∞
∑
ϕ X (t0 ) = 1, | t0 |> 0 then
k = −∞
P( X =
2πk
) = 1.
t0
(b) (5 points) Prove that if there is a sequence of numbers t n → 0 so that
ϕ X (t n ) = 1 for each n, then P( X = 0) = 1 .
(c) (5 points) Assume that X n − Yn ⇒ 0 where for each n { X n , Yn } are i.i.d.
Assume also that there is t n → 0, | t n |> 0 with
ϕ X n (t n ) = 1 for each n. Is it
necessarily true that X n ⇒ 0 ? Prove or produce a counter example.