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
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 roll
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 he
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
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
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
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 d
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
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 a
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
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 corr
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
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
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
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
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 Chapte
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
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 mart
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 corres
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 P