Math 6242 HW 10
Due Tues., Mar. 29
Let S be a countable state space for a Markov chain.
1. (Durrett, 6.6.1) Show that if S is finite and (Xn ) is irreducible and aperiodic, then
there is an m such that pm (x, y) > 0 for all x, y.
2. (Durrett, 6.6.2) Suppo
Math 6242 HW 7
Due Tues., Mar. 1
1. (Durrett, 6.2.2) Suppose S = cfw_1, 2, 3 and
.1 0 .9
P = .7 .3 0 .
0 .4 .6
2
3
Compute P1,2
and P2,3
by considering the different ways to get from 1 to 2 in two steps
and 2 to 3 in three steps.
2. (Durrett, 6.2.3) Suppo
Math 6242 HW 4
Due Tues., Feb. 9
For this assignment, let (, , P) be a probability space and (Fn ) a filtration.
1. Let (Xi )iI be a collection of random variables such that |Xi | Y for all i and some
Y L1 . Show that (Xi ) is uniformly integrable.
2. Sho
Math 6242 HW 2
Due Tues., Jan. 26
For this assignment, let (, , P) be a probability space and (Fn ) a filtration.
1. (Durrett, 5.2.2) A function f : Rd R is called superharmonic if it has continuous
derivatives of order 2, and satisfies
2
2
+ + 2 f (x) 0
Math 6242 HW 8
Due Tues., Mar. 8
1. Let (Xn ) be a Markov chain on a countable state space with initial distribution given
by some stationary distribution . Show that for a fixed n, the sequence
Xn , Xn1 , Xn2 , . . . , X0
is also a Markov chain and compu
Math 6242 HW 12
Due Tues., Apr. 19
Let (Bt ) be a standard Brownian motion.
1. Show that almost surely, there is no nonempty interval I [0, ) such that Bt is
non-increasing or non-decreasing on I. Conclude that almost surely, the set of t at
which Bt has
Math 6242 HW 1
Due Tues., Jan. 19
For this assignment, let (, , P) be a probability space and F a sub-sigma-algebra.
1. (Conditional dominated onvergence) Let (Xn ) and X be a sequence of random variables
such that for some integrable random variable Y ,
Math 6242 HW 11
Due Tues., Apr. 12
Let (Bt ) be a standard Brownian motion.
1. Show that the set of right-continuous functions is not measurable in the product Borel
sigma-algebra of R[0,) . What about the set of functions which are continuous at
Lebesgue
Math 6242 HW 3
Due Tues., Feb. 2
For this assignment, let (, , P) be a probability space and (Fn ) a filtration.
1. (Durrett, 5.3.2) Give an example of a martingale (Xn ) with supn |Xn | < almost
surely and P(Xn = a i.o.) = 1 for a = 1, 0, 1.
2. (Durrett,
Math 6242 HW 9
Due Tues., Mar. 15
Let S be a countable state space for a Markov chain.
1. (Durrett, 6.2.6) Suppose two urns, which we will call left and right, have m balls each.
b (which we will assume is m) balls are black, and 2m b are white. At each t