MAT 902 STOCHASTIK II
HOMEWORK 1
Problem 1. Let (Ω, A, P ) be a probability space.
a) A family L of subsets of Ω is called a λ-system if it satisfies i) Ω ∈ L, ii) if A ∈ L, then
AC ∈ L iii) for each sequence Ai , i ≥ 1, of pair-wise disjoint elements from L, one has
(∪i≥1 Ai ) ∈ L. Show that a λ-system can be also defined as a family D of subsets of Ω
satisfying
i) Ω ∈ D,
ii) if B, B 0 ∈ D and B 0 ⊂ B, then B \ B 0 ∈ D,
iii) if Bi ∈ D and Bi ⊂ Bi+1 , then ∪i≥1 Bi ∈ D.
b) A family C of subsets of Ω is a π-system if it is closed under intersection, that is, if
A, B ∈ C then A ∩ B ∈ C. Show that a λ-system L is a σ-algebra if and only if it is a
π-system.
c) Dynkin’s π − λ theorem states that if C is a π -system and L is a λ-system that contains
C, then σ(C) ⊂ L.
Now suppose C1 , . . . , Cn ⊆ A are π-systems with Ω ∈ Ci and
∀C1 ∈ C1 , . . . , Cn ∈ Cn , P [C1 ∩ · · · ∩ Cn ] = P [C1 ] · · · P [Cn ].
Show that the σ-algebras σ(C1 ), . . . , σ(Cn ) are independent.
Hint: Consider the family D1 of all D ∈ A such that P [D ∩ C2 ∩ · · · ∩ Cn ] =
P [D]P [C2 ] · · · P [Cn ]. Show that D1 is a λ-system and contains C1 . Continue the process
inductively.
d) Suppose Fi,j , 1 ≤ i ≤ n, 1 ≤ j ≤ m(i) are independent σ-algebras and let Gi =
m(i)
σ(∪j=1 Fi,j ). Show that the σ-algebras G1 , . . . , Gn are also independent.
Problem 2. Let X and Y be independent random variables taking values in R, with X a
standard normal random variable (X ∼ N (0, 1)), and Y with density function
(
2
te−t /2 if t ≥ 0
P (t) :=
0
otherwise.
Find the density function of the random variable Z := X · Y .
Problem 3. Let H be a Hilbert space, and u1 , u2 , ..., un ∈ H. Let X1 , ..., Xn be i.i.d. random
variables such that for each i, Xi takes the value 1 with probability 1/2 and the value −1 with
probability 1/2. Prove that
n
n
X
X
2
Ek
Xi ui k =
kui k2 .
i=1
i=1