310B: HOMEWORK SOLUTIONS 2015
405
Homework 5
Solution. 5.2.5
(m)
Let Yn = E(Xm )+ |Fn ), m n. By Jensens inequality and the tower property
of conditional expectation
(m+1)
(m)
Yn
= E(E(Xm+1 )+ |Fm )|Fn ) E(E(Xm+1 |Fm )+ |Fn ) = Yn
(m)
is non-decreasing fo
Statistics 310B, Winter 2015. Solutions, Homework 8.
(1) (a) Let T () = 1 . It preserves P and clearly there are no invariant sets
other than and . However T 2 is the identity and is not ergodic (cfw_1
and cfw_0 are invariant).
(b) Consider the set B = cf
Statistics 310B, Winter 2015. Homework 8. Due March 4
Read all that was not done in class out of Durretts 7.17.2.
1. (a) For = cfw_0, 1, F = 2 and P(cfw_0) = P(cfw_1) = 1/2, nd an ergodic measure
preserving T such that T 2 is not ergodic.
(b) Let T on (,
310B: HOMEWORK SOLUTIONS 2015
395
Homework 3
Solution. 5.1.13
For any n the event cfw_ n = n cfw_Xk B is clearly in Fn = (Xk , k n).
k=0
Hence, is a stopping time with respect to this ltration. In contrast, cfw_ n =
/
k>n cfw_Xk B which for independent, n
310B: HOMEWORK SETS 2015
389
310b: Homework Sets 2015
Homework rules: Arguments should be precise, concise and correct. We expect
rigorous and complete proofs. Answers should be clean and legible. Please use a
separate sheet of paper for each problem (eas
310B: HOMEWORK SOLUTIONS 2015
411
Homework 6
Solution. 5.3.39
Set > 0 and let = e 1.
(a) Obviously, Nn is measurable on Fn . By our assumptions about the L2
martingale (Mn , Fn ), part (a) of Exercise 1.4.40 applies for the law of
Y = (Mn+1 Mn ) condition
310B: HOMEWORK SOLUTIONS 2015
417
Homework 7
Solution. 6.1.14
(a) Using the notations X n = (X1 , . . . , Xn ) and xn = (x1 , . . . , xn ), observe
that for any n and xn cfw_1, 1n,
P(X n = xn |) = t (1 )nt ,
where t = t(xn ) =
n
i=1
1xi =1 . Thus, conditi
310B: HOMEWORK SOLUTIONS 2015
389
Homework 2
Solution. 4.3.11
In the proof of Theorem 4.1.2 we use the Radon-Nikodym theorem only for showing
the existence of C.E. in case X L1 (, F , P) and X 0.
So, assuming that X is non-negative and integrable we proce
310b: Homework Solutions 2015
Homework 1
Solution. 4.1.3
We need to show that E(XIA ) = E(Y IA ), for all A G = (P). Let L = cfw_A
F : E(XIA ) = E(Y IA ). The assumption implies P L. By Dynkins
theorem, it suces to show that L is -system, which we proce
310B: HOMEWORK SOLUTIONS 2015
425
Exam practice.
Solution. 5.2.16
(a) Fixing n and t, s 0 let = mincfw_k 1 : |Sk | t + s. Note that if
n and |Sn | < t, then necessarily |Sn S | > s. Consequently, with
a stopping time for the canonical ltration Gm = (Sk
400
310B: HOMEWORK SOLUTIONS 2015
Homework 4
Solution. 5.3.3
First recall Example 5.1.23 that (Xn )+ is a submartingale. This implies that the
sequence E(Xn )+ is non-decreasing, hence converges and consequently, that (d) is
equivalent to (e). Clearly (a)
Stat 310B/Math 230B Theory of Probability
Practice Midterm
Andrea Montanari
Due on 2/4/2014
Solutions should be complete and concisely written. Please, use a separate booklet for each problem.
You have 3 hours but you are not required to solve all the pro