P ROBABILITY I
9TH E XERCISE SHEET - D UE F RIDAY, J UNE 03, 2016
P ROFESSOR
M ONITOR
: Hubert Lacoin
: Enrique Chávez Sarmiento
Friday, May 27, 2016
The solutions of the star-marked exercises should be presented until the deadline. Among the
others exercises, choose two (and only two) and attach their solutions to the previous ones.
1. Let (Ω, F , P) be a probability space. Let P be a π-system generating the sub-σ-algebra G , and
e = E[ X |G]
suppose that Ω is a countable union of sets in P . Prove that in case X be integrable, X
if and only if
Z
Z
e dP =
X
X dP, ∀ E ∈ P .
E
E
(?) 2. Show that the independence of σ( X ) and F implies that E[ X |F ] = E[ X ].
3. 1. Generalize Markov’s inequality:
E[1{|X |≥α} |F ] ≤
E[| X ||F ]
,
α
∀α > 0.
2. Similarly generalize Chevishev’s and Hölder’s inequalities.
4. Assume that E[ X 2 ], E[Y 2 ] < ∞. Show the equality
E XE[Y |F ] = E YE[ X |F ] .
(?) 5. Consider a probability space (Ω, F , P). Let
V = { X : Ω → R : X is a random variable such that E[ X 2 ] < ∞}.
We define
L2 = L2 (P) := V/ ∼,
where X ∼ Y if and only if X = Y a.s.
and [ X ] denotes the equivalence class of X ∈ V.
1. Prove that L2 is a Hilbert space with the inner product h[ X ], [Y ]i := E[ XY ].
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2. Given a σ-algebra G ⊂ F , let M be the subspace of elements
[ X ] of L such that X is G 2
measurable. Show that P : L → M defined by P([ X ]) = E[ X |G] is the perpendicular
projection on M.
6. In a probability space (Ω, F , P), we say that a sequence of events A1 , A2 , . . . is mixing (or Pmixing) with constant α if
lim P[ An ∩ E] = αP[ E], ∀ E ∈ F .
(1)
n→∞
Note that in particular limn P[ An ] = α.
1
1. Prove that ( An )n∈N is mixing with constant α if and only if
lim E[1 An X ] = αEX,
n→∞
for all integrable random variables X.
2. Let P be a π-system. Suppose that (1) holds for all E ∈ P ∪ {Ω}, and An ∈ σ(P ) for all
n ∈ N. Show that ( An )n∈N is mixing.
e is a probability absolutely continuous with respect to P. Prove that if
3. Suppose that P
e
( An )n∈N is P-mixing, then it is also P-mixing.
4. Let [0, 1], B([0, 1]), P be a probability space, where P has uniform distribution. Prove that
there exists a mixing sequence of events with constant α = 1/2. Is this true for α = 1/3 or
α = 2/3? Try to generalize.
5. Prove that there is no mixing sequence of events with constant α ∈ (0, 1) over a probability
space (N, F , P).
1 Definition. Let (Ω, F , P) be a probability space. We say that the family of sub-σ-algebras
{Fn }n∈N is a filtration on N if Fn ⊂ Fn+1 for all n ∈ N.
2 Definition (Stopping time). Given a probability space (Ω, F , P), let {Fn }n∈N be a filtration on
N. A random variable T : Ω → N is called stopping time of {Fn }n∈N if
{ T ≤ n} ∈ Fn ,
∀n ∈ N.
7. Let (Ω, F , P) be a probability space with a filtration {Fn }n∈N .
1. Prove that T is an stopping time if, and only if, 1{T =n} is Fn -measurable for all n ∈ N.
2. Suppose that the sequence of random variables ( Xn )n∈N is adapted, i.e. Xn is Fn -measurable
for all n ∈ N. For a fixed A ∈ B(R) assume that
T := inf{n ∈ N : Xn ∈ A},
is finite. Prove that T is an stopping time.
3. Let T be an stopping time. Prove that
FT := { A ∈ F : A ∩ { T ≤ t} ∈ Fn for each n ∈ N}
is a σ-algebra.
4. Suppose that S, T are stopping times with S ≤ T. Prove that S ∧ T and S ∨ T are stopping
times.
5. Let T be an stopping time. Prove that a random variable X is F T -measurable if and only if
X1{T =n} is Fn -measurable for all n ∈ N.
8 (continuation). On the conditions of Exercise 7 we assume that S and T are stopping times
1. Suppose that ( Xn )n∈N is adapted, we define XT (ω ) := XT (ω ) (ω ). Prove that XT is F T measurable. Moreover, show that every F T -measurable random variable can be written as
XT for some adapted sequence of random variables ( Xn )n∈N .
Hint: Observe that XT 1{T =n} = Xn 1{T =n} .
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2. Suppose that X is FS -measurable. Prove that the variables
X1{S≥T } ,
X1{S=T } ,
X1{S<T } ,
are FS∧T -measurable.
3. Let A ∈ FS ∩ FT . Using the previous item prove that A ∩ {S ≤ T } ∈ FS∧T and A ∩ { T ≤
S} ∈ FS∧T . Deduce FS ∩ F T ⊂ FS∧T , and conclude the equality FS ∩ F T = FS∧T .
9 (continuation). We are in the context of Exercise 8. The purpose of this exercise is to prove that
E E[ X |FT ]|FS = E[ X |FS∧T ],
(2)
for all integrable random variables X. For convenience we will write just ER [ · ] instead of E[ · |F R ]
for every stopping time R.
1. Show that (2) is equivalent to ES [ET X ] = ES∧T [ET X ].
2. Here and henceforth we denote Y := ET X. Prove that
E[ ZY ] = E[ ZES∧T Y ],
for all positive and FS -measurable Z,
implies (2).
3. Let Z be like before. Show that
E[ Z1{S≤T } Y ] = E[ Z1{S≤T } ES∧T Y ].
Hint: Use Exercise 8.
4. Let Z be like before. Observe that Y1{T <S} is FS∧T -measurable and deduce
E[ ZY1{T <S} ] = E[ Z1{T <S} ES∧T Y ].
Hint: Use Exercise 8.
5. Conclude.
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