# Homework

## Homework

• Homework set 1: Due Jan 26
Problems (Not exercises)
1.2: 1,6,13
1.3: 6,7,8a,9,11
1.4: 3,5
1.5: 1,7
Solutions.

• Homework set 2: Due Feb 2
Problems (Not exercises)
2.1: 1a,2,4,8
2.3: 1,3,4 Solutions.

• Homework set 3: Due Feb 9
Problems (Not exercises)
3.1: 2,4
3.2: 1,2,4,5 Solutions.

• Homework set 4: Due Feb 18
Problem 1. Consider the sequence defined recursively by $a_1=3, a_2=7, a_n=a_{n-1}+a_{n-2},n\geq 3$. Find a closed form formula for $a_n$. Note that if you do this for $a_1=a_2=1$ you get the Fibonacci sequence.
Problem 2. Let $X_n$ be a Markov Chain with stationary transition probabilities. a) Let $i$ be any state. Prove that $P(X_n=i \text{ for infinitely many indices } n|X_0=j)=0\forall j$ if and only if $P(X_n=i \text{ for infinitely many indices } n|X_0=i)=0$. \ b) Prove that $P(X_n=i \text{ for infinitely many indices } n|X_0=i)=0$ if and only if $P(\exists n>0: X_n=i|X_0=i)\neq 1$.
Make sure you give rigorous arguments using the Markov Property and note some heuristics.
Exercise 3.4.8
Problems
3.4: 1,2,14,18
Solutions.

• Homework set 5: Due Feb 25
Exercises
3.5.6, 3.6.1
Problems
3.4.12,3.4.17,3.5.5 (See page 72 for the definition of a martingale),3.7.1
Solutions.

• Midterm exam
Solutions.

• Homework set 6: Due Mar 18
Problem 1. Let $a_n$ be a sequence such that $\lim_{n\rightarrow\infty}a_n=a$. Prove that $\lim_{m\rightarrow\infty}\frac{a_1+\dots+a_m}{m}=a$ as well.
Exercises
3.8.1, 3.8.3(only do n=4)
Problems
3.8.2,3.8.3
Solutions.

• Homework set 7: Due Mar 25
Exercises
4.1.1
Problems
3.9.1,3.9.8,4.1.1,4.1.4,4.1.6,4.1.13
Solutions.

• Homework set 8: Due Apr 1
Exercises
4.3.1,4.3.4
Problems
4.3.2,4.4.4,4.4.5,5.1.1,5.1.12
Solutions.

• Homework set 9: Due Apr 15
Problem 1. Let $N$ be a Poisson point process of rate $\lambda$. Let $W_k$ be the time of the $k$’th event. Let $Y_1,Y_2,\dots$ be independent random variables with density $g$. Let $0<t_1<t_2$ and let $h_1<h_2<\dots<h_{k+1}$. Define $N_m$ to be the number of points $(W_i,Y_i)$ in the rectangle $(t_1,t_2]\times(h_m,h_{m+1}]$. Prove that $N_1,\dots,N_k$ are independent and that $N_m$ is a Poisson random variable with rate $\iint_A \lambda g(y)dy dt$.
Exercises
5.3.4
Problems
5.2.4,5.2.11,5.3.8,5.4.7
Solutions.

• Homework set 10: Due Apr 27 (NOTE: This is due on Wednesday, as that’s the last day of classes!)
Exercises
5.6.1,5.6.4,6.1.3
Problems
5.5.4,5.6.1,6.1.1,6.1.8,6.2.1,6.4.2
Solutions.

• Final exam
Solutions.