Preliminary Exam: Probability
Friday, August 28, 2009
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 7 problems. The several steps that problems 3-7 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. (5 points) Let X and Y be independent random variables and let
f : R → R and g : R → R be measurable functions. Prove that if
E | f ( X ) − g (Y ) |< ∞ then E | f ( X ) |< ∞ and E | g (Y ) |< ∞ .
Problem 2. (7 points) Let { X n , n ≥ 0} be a sequence of random variables that are
integer-valued. Assume that for each k ∈ Z
P ( X n = k ) ⎯n⎯
⎯→ P ( X 0 = k ) .
→∞
Prove that
⎯→ 0 .
∑ | P( X n = k ) − P( X 0 = k ) | ⎯n⎯
→∞
k ∈Z
σ − algebras. Let
{S k , k = 0,..., n} be an L2 martingale with respect to {Fk } with S 0 = 0 . Let
λ > 0 and define: τ = min{0 ≤ k ≤ n : | S k |> λ}
(τ = n if no k with | S k |> λ exists)
Problem 3. Let Fk , k = 0,..., n be an increasing sequence of
k
a. (5 points) Denote X k = S k − S k −1 , 1 ≤ k ≤ n . Prove that Yk = ∑ Si −1 X i
i =1
is a MG. What is E (Yτ ) ?
b. (5 points) Prove that E ( Sτ −1Sτ ) ≤ λE (| S n |) .
c. (3 points) Assume the algebraic identity :
k −1
k
i =1
i =1
S k2−1 + ∑ X i2 = 2 Sk −1S k − 2∑ Si −1 X i , 2 ≤ k ≤ n ,
and prove that
E ( Sτ2−1
+
τ −1
∑ X i2 ) ≤ 2λE (| S n |) .
i =1
Problem 4. Let { X n , n ≥ 0} be a sequence of random variables taking values in [0, 1].
We set Fn = σ { X k , k = 0,1,..., n} . Assume that X 0 = a a.s., where a ∈ [0,1] is a
constant, and
P( X n +1 =
Xn
1+ X n
| Fn ) = 1 − X n , P( X n +1 =
| Fn ) = X n
2
2
a. (7 points) Prove that { X n , n ≥ 0} is a martingale which converges a.s. and in L to a
random variable Y .
2
b. (4 points) Show that E[ X n +1 − X n ) ] = E[ X n (1 − X n )] / 4
2
c. (7 points) Compute E[Y (1 − Y )] and determine the distribution of Y .
Problem 5. Let ε , U and W be independent random variables with:
(i)
P(ε = 1) = P(ε = −1) = 1 / 2 ,
α
(ii)
0 < E | W | < ∞ where α > 0 is a constant, and
(iii)
U is uniformly distributed on (0, 1).
We define
Y = ε ⋅ U −1 / α ⋅ W
a. (5 points) Prove that Y is symmetric, unbounded with P (Y = y ) = 0, | y |> 0 .
b. (5 points) Prove that
lim λα P(| Y | > λ ) = E | W |α
λ →∞
c. (5 points) Define {an } , a positive and increasing sequence of constants with
an → ∞ , such that for each λ > 0 we have:
lim n ⋅ P(| Y | > an λ ) = λ−α
n→∞
Problem 6. Let Wt , t ≥ 0 be a standard Brownian motion. Let
τ = inf{t : Wt ∉ (−1, 2)}.
a. (5 points) Prove directly that X t = Wt − 6tWt + 3t is a martingale with
respect to {Ft = σ (Ws , 0 ≤ s ≤ t )} , namely show by using the basic definition of
Brownian motion that E ( X t − X s | Fs ) = 0, 0 ≤ s ≤ t .
4
2
2
b. (6 points) Find E (τ ) and E (τ ⋅ Wτ ) . Justify each step in your calculations.
Hint: you may use without proof that
Wt 3 − 3tWt is a martingale.
c. (5 points) Use the values of E (τ ) and E (τ ⋅ Wτ ) to develop a system of 2
equations for
A = E (τ | Wτ = −1) and B = E (τ | Wτ = 2) . Find A and B by solving the
system.
d. (5 points) Find E (τ ⋅ Wτ ) and E (τ ) .
2
2
Problem 7. Let { X n , n ≥ 1} be a sequence of independent and symmetric random
variables. Let S n =
n
∑ X k , Ym = S 2
m +1
k =1
a. (5 points) Prove that if
ε > 0.
− S 2 m , and Ym* = mmaxm +1 | S k − S 2m | .
2 <k ≤2
Sn
→ 0 a.s. then
n
∞
∑ P(Ym > 2mε ) < ∞
∞
b. (5 points) Prove that
∑ P(Ym > 2mε ) < ∞
for all
ε > 0 if and only if
m=0
Ym*
→ 0 a.s.
2m
∞
c. (7 points) Prove that if
∑ P(Ym > 2mε ) < ∞
for all
ε > 0 then
m=0
S2 m
2
m
→ 0 a.s. (Hint: Try to express
∞
d. (4 points) Show that if
2
m
by using {
Yk
})
2k
∑ P(Ym > 2mε ) < ∞ for all ε > 0 then
m=0
Sn
→ 0 a.s.
n
S 2m
for all
m=0