Solutions to homework 13
Statistics 205B: Spring 2008
1. (Problem 4.1 from section 7.4 in Durrett) (a) Generalize the proof of 7.4.6 to conclude that if u < v a then P0 (Ta < t, u < Bt < v) = P0 (2a - v < Bt < 2a - u). (b) Let Mt = max0st Bs . Use (a) to
Solutions to Homework 12
Statistics 205B: Spring 2008 1. (Problem 5.1 from section 7.5 in Durrett) Let T = infcfw_t : Bt (-a, a). Prove that / E0 exp(-T ) = 1/ cosh(a 2)
by showing that exp(- 2 t/2) cosh(Bt ) is a martingale. Solution : Note that Xt = exp
Solution to homework 11
Statistics 205B: Spring 2008
1. Use Kolmogorov 0 - 1 law in order to prove that if B is a BM then infcfw_t > 0 : Bt > 0 = 0 a.s. Solution: Enough to how that = infcfw_2-n : B2-n > 0 = 0. First note that P( 2-n ) P(B2-n > 0) = 1/2 f
Solution to homework 10
Statistics 205B: Spring 2008 1. (Problem 1.3 from section 7.1 in Durrett) Fix t and let m,n = B(tm2-n ) - B(t(m - 1)2-n ). Compute 2 E
m2n
2 - t m,n
and use Borel-Cantelli Lemma to conclude that
Solution: For fixed n, clearly m,n
Solutions to homework 9
Statistics 205B: Spring 2008 1. Let X and Y be two Rd valued random variables with E[X] = E[Y] = 0 and Cov[X] = Cov[Y] finite. Let X1 , X2 , . . . be independent copies of X and Y1 , Y2 , . . . independent copies of Y. Let Un = n-1
Solutions to homework 8
Statistics 205B: Spring 2008
1. (Problem 1.12 from section 3.1 in Durrett) Let X1 , X2 , X3 , . . . be i.i.d. uniform on (0, 1), let Sn = X1 + X2 + + Xn , and T = infcfw_n : Sn > 1. Show that P(T > n) = 1/n!, so E T = e and EST = e
Solutions to homework 7
Statistics 205B: Spring 2008 1. Reading exercise: Read (and understand) Example 6.1.5, Theorem 6.1.2 and Theorem 6.1.3 in Durrett. Give one instance where we have used Theorem 6.1.2 and one where we have used Theorem 6.1.3. Solutio
Solution to Homework 6
Statistics 205B: Spring 2008
1. (Problem 1.2 from section 6.1 in Durrett) (a) Let A be any set and let B = -n (A) . Show that -1 (B) B. n=0 (b) Let B be any set with -1 (B) B and let C = -n (B). Show that -1 (C) = C. n=0 (c) Show th
Solution to Homework 5
Statistics 205B: Spring 2008 1. Suppose T is a strong stationary stopping time so that for every value of x if X0 = x then conditioned on T = t, the distribution of XT is given by the stationary distribution . Suppose further that f
MARKOV CHAINS
What I will talk about in class is pretty close to Durrett Chapter 5 sections 1-5. We stick to the countable state case, except where otherwise mentioned. Lecture 7. We can regard (p(i, j) as defining a (maybe infinite) matrix P. Then a basi
Homework 13
Statistics 205B: Spring 2008 Due on May 1, 2008 1. (Problem 4.1 from section 7.4 in Durrett) (a) Generalize the proof of 7.4.6 to conclude that if u < v a then P0 (Ta < t, u < Bt < v) = P0 (2a - v < Bt < 2a - u). (b) Let Mt = max0st Bs . Use (
Homework 10
Statistics 205B: Spring 2008 Due on April 10, 2008
1. (Problem 1.3 from section 7.1 in Durrett) Fix t and let m,n = B(tm2-n ) - B(t(m - 1)2-n ). Compute
m2n
E
2 - t m,n
m2n
2
and use Borel-Cantelli Lemma to conclude that 2. (Problem 1 from c