Math 136 - Stochastic Processes Homework Set 6, Autumn 2009, Due: Nov 4 1. Exercise 4.1.5. Suppose (Xn , Fn ) is a martingale. Show that then cfw_Xn is also a martingale with respect to its canonical ltration and that a.s. E [Xl |Fn ] = Xn for all l > n.
Stats 219 - Stochastic Processes Homework Set 7, Fall 2009 1. Exercise 4.2.4 (a) This can be easily proved by replacing t + h with t and t with s in the identity as given in hints. Note that At is a deterministic process and hence comes out of the conditi
Stochastic Processes Amir Dembo (Revised by Kevin Ross) September 25, 2008
E-mail address: [email protected], [email protected] Department of Statistics, Stanford University, Stanford, CA 94305.
Contents
Preface Chapter 1. Probability, measure
Math136/Stat219 Fall 2009 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. Y
Stat219 / Math 136 - Stochastic Processes Homework Set 1, Fall 2009. Due: Wednesday, September 1 1. Exercise 1.1.3. Let (, F , IP) be a probability space and A, B, Ai events in F . Prove the following properties of IP. (a) Monotonicity. If A B then IP(A)
Math 136 - Stochastic Processes
Homework Set 6, Autumn 2013, Due: November 6
1. Exercise 4.1.6 Provide an example of a probability space (, F, P), a ltration cfw_Fn and a stochastic
process cfw_Xn adapted to cfw_Fn such that:
(a) cfw_Xn is a martingal
Math 136 - Stochastic Processes
Homework Set 4, Autumn 2013, Due: October 23
1. Exercise 2.3.19. Suppose that X and Y are square integrable random variables.
(a) Show that if E(X|Y ) = E(X) then X and Y are uncorrelated.
ANS: By the tower property and tak
Math 136 - Stochastic Processes
Homework Set 7, Autumn 2013, Due: November 13
1. Exercise 4.3.4. Show that the rst hitting time () = mincfw_k 0 : Xk () B of a Borel set B R by
a sequence cfw_Xk , is a stopping time for the canonical ltration Fn = (Xk , k
Math 136 - Stochastic Processes
Homework Set 5, Autumn 2013, Due: October 30
1. Exercise 3.2.21. Consider the random variables Sk of Example 1.4.13.
(a) Applying Proposition 3.2.6 verify that the corresponding characteristic functions are
S () = [cos(/ k)
Math136/Stat219 Fall 2009 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. Y
Conditional expectation
Jason Swanson April 17, 2009
1
Conditioning on -algebras
P (B A) , P (A)
Let (, F , P ) be a probability space and let A F with P (A) > 0. Dene Q(B ) = P (B | A) = for all B F .
It is easy to to check that Q is a probability measur
Stat 219 - Stochastic Processes Homework Set 2, Fall 2009, Due: October 7 1. Exercise 1.2.22. Write (, F , P ) for a random experiment whose outcome is a recording of the results of n independent rolls of a balanced six-sided dice(including their order).
Stat 219 - Stochastic Processes Homework Set 3, Autumn 2009 1. Exercise 1.4.31. Prove Proposition 1.4.3 using the following steps. (a) Verify that the identity (1.4.1) holds for indicator functions g (x) = I B (x) for B B . ANS: Let B B be an arbitrary Bo
Math 136 - Stochastic Processes Homework Set 4, Fall 2009 1. Exercise 2.3.8. The left hand side is the smallest distance from G2 , while by embeddedness, right hand side is a distance from G2 , and hence the inequality holds.(no need for algebra) 2. Exerc
Math 136 - Stochastic Processes Homework Set 5, Fall 2009, Due: October 28 1. Exercise 3.2.21. Consider the random variables Sk of Example 1.4.13. (a) Applying Proposition 3.2.6 verify that the corresponding characteristic functions are Sk () = [cos(/ k )
Stat 219 - Stochastic Processes Homework Set 8, Fall 2009, Due: November 18th 1. Exercise 5.2.5. Show that E(, ) = by applying Doobs optional stopping theorem for the uniformly integrable stopped martingale Wt2 , t , . ANS: We have seen en-route to (5.2.2
STATS 219 - Stochastic Processes HW 9, Autumn 2009 1. Exercise 4.6.8. Suppose cfw_Zn is a branching process with P(N = 1) < 1 and Z 0 = 1. Show that P( lim Zn = ) = 1 pex ,
n
rst in case m 1, then in case P(N = 0) = 0 and nally using the preceding exerci
STAT 219/ MATH 136 Fall 2009 Midterm Solutions
Friday, October 23
Write your name and sign the Honor code in the blue books provided.
This is a closed note and closed book exam. You have 50 minutes to solve all THREE questions, each worth points as marked
Stochastic Processes Amir Dembo (revised by Kevin Ross) April 8, 2008
E-mail address : [email protected] Department of Statistics, Stanford University, Stanford, CA 94305.
Contents
Preface Chapter 1. Probability, measure and integration 1.1. Probabil
Use of the Perron-Frobenius Theorem
Theorem
If P be a stochastic matrix with positive entries
anda unique invariant
~
.
n
probability distribution ~ , then limn P = .
~
Doron Shahar
Proof of the Perron-Frobenius Theorem
May 1, 2014
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Use of the P