In this course we discuss the foundations of stochastic processes: everything you wanted to know about random processes but you were afraid to ask. As a result, we talk every now and again about some advanced notions in probability. For instance we start by Sigma algebra, measurable functions, and Lebesgue integral. We then look at Poisson processes, on of the simplest, yet interesting random models. We continue the course with what we mean by the convergence of random variables. Later, we look at renewal processes, a more general form of counting processes. The theory of renewal processes will equip us with many interesting and useful techniques which will become extremely handy when we next talk about Markov chains. Finally we discuss Martingales in quite detail. If time permits we will talk about more advanced topics such as graphical models.

Time and Place: There will be two lectures per week - Monday and Wednesday in WLH 208 at 1:00-2:15

Lecturer: Amin Karbasi (amin.karbasi@yale.edu), office hours: Monday 4:00-5:30.

TAs:

Lin Chen (lin.chen@yale.edu), Thursday 4-6pm, 17 Hillhouse, room 333

Xiaoqian (Dana) Yang (xiaoqian.yang@yale.edu), Tuesday 4-6pm, 24 Hillhouse, Stat classroom

Mehraveh Salehi (mehraveh.salehi@yale.edu), Wednesday 4-6pm, 17 Hillhouse, room 333

Grading: The grade will be based on 5 sets of homework (each counts towards 20%). Class participation (e.g., interaction during the lecture, asking good questions) can also influence your grade.

Collaboration: In case of problem sets, collaboration is also allowed, but: (1) the writeup of all the solutions has to be prepared independently (in particular, you should understand and be able to explain everything that is written in your solution); (2) in your writeup, you should include the name of your collaborator, 3) never, ever copy and past.

Problem sets: For every problem set, you have exactly one week to return.

Books: For advanced probability and Martingale we use lecture notes of Olivier Leveque.

For Poisson Processes we use Introduction to Stochastic Processes , Renewal Processes and Markov Chains we use Stochastic Processes: Theory for Applications.

Other references: Chang’s book.

Prerequisites: I assume you know basic probability concepts (you should have taken STAT 241 or an equivalent course). To refresh your mind, or when you are in doubt, you can look at Introduction to Probability, Statistics, and Random Processes.

### Homework 1

Exceptionally, For the first homework you have time until Friday 19th, 4pm to return your solutions for HOMEWORK1

### Recitation 3

Problems for the third recitation

### Homework 2

For this homework, you have time until April 1st to make sure that you can all enjoy the break. Check the problems.

### Homework 3

For this homework, you have time until April 13th. Check the problems.

### Homework 4

For this homework, you have time until April 26th, 5pm. Check the problems.

### Final exam

You have time until May 10th, 7pm to return your solutions. Check the final exam.