ENEE 620: Random Processes

in Communications and Control
Fall 2018

Instructor : Alexander Barg, Professor
Department of Electrical and Computer Engineering/Institute for Systems Research
Office: 2361 A.V.Williams Building. E-mail abarg at  umd  dot edu
Office hours: Wed. 3:00-4:30, or send me an email

Class times:
Lectures: Tuesday, Thursday 12:30-1:45pm CSI1122
Discussion sessions:
Friday 11:00am-11:50am CSI2118 (Section 0101)
Friday 11:00-11:50 CSI1121 (Section 0102)

TAs:
Section 0101: Aneesh Raghavan, Email: raghava@terpmail.umd.edu
Office hours: Wednesday 1:00-2:30 AVW1301

Section 0102: Zitan Chen, Email: chenztan@gmail com
Office hours: Monday 2:30 - 4:00 AVW2456

Course Homepage: http://www.ece.umd.edu/~abarg/620

Grading: Several (5-6) home assignments (20%), midterm1(25%), midterm2 (25%), final (30%).
Exams are closed-book, no calculators or other electronic devices. You can bring one letter-size sheet of notes to any exam, you may write on both sides.

Textbooks (recommended):
[H]: B. Hajek, Random Processes for Engineers, Cambridge University Press, 2015. Web link
[G]: R.G. Gallager, Stochastic Processes: Theory and Applications, Cambrdige University Press, 2014 Web link

Home assignments: Are due in a week upon being assigned (unless announced otherwise)

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Lect. #TopicsTextbooksReadingHWSolutions
1 (8/28) Course description. What is probability, by example:
Borel's Normal Numbers
[G], Sec.1.1, 1.8[H] 11.1, 11.2
Billingsley [B] Sec.1.1
2 (8/30) Random points in (0,1] and normal numbers. WLLN and SLLN. Axioms of probability, algebras and σ-algebras [H], Sec.1.1, 1.2; [G], Sec.1.2 [H], 11.3,11.4,11.5.1
[G], Sec.1.7; [V],Ch.V
Prof. Tao's SLLN blog
HW1Solutions
3 (9/4) Continuity of probability, Borel-Cantelli lemma [H], Sec.1.1[V], Sec.IV.4
4 (9/6) Random Variables (RVs), Distribution functions, PDFs [H], Sec.1.3[V], Sec.XI.1-2; XII.1-3
[B], Sec.5
5 (9/11) Expectation of an RV (motivation and approach to a general definition) [H], Sec.1.5[V], Ch.XIII.3-5
[L1], Sec. 4
[B], Sec.5 and 21
6 (9/13) Jointly distributed RVs, functions of RVs [H], Sec.1.8HW2
Practice problems
7 (9/18) Modes of convergence of RVs
Difference between convergence in probability and almost surely
[H], Sec.2.1
[G], pp.45-51
[V], Sec.XII.5
8 (9/20) Modes of convergence. Covergence in distribution [H], Sec.2.1
9 (9/25) Cauchy sequences of RVs; criteria for convergence [H], Sec. 2.2
10 (9/27) Laws of Large Numbers [H], Sec. 2.3
[G], Sec. 1.7
[L1], Sec.5.5
11 (10/2) Random Processes. The Gambler's Ruin Problem
12 (10/4) Counting processes. The Poisson process
13 (10/9) The Poisson Process [Ly], Ch. 2
(10/11) Midterm 1
14 (10/16) Discrete-time Markov chains: Basic properties
15 (10/18) Discrete-time Markov chains
16 (10/23) Continuous-time Markov chains
17 (10/25) Gaussian processes: Jointly Gaussian RVs and Gaussian vectors [G]
18 (10/20) Gaussian processes: Stationarity, orthogonal expansions
19 (11/1) White Gaussian noise. Wiener process
20 (11/6) Random walks
21 (11/8) Random walks: threshold crossing, Wald's identity
22 (11/13) Midterm 2 (tentative)
23 (11/15) Conditional expectation
Filtrations; orthogonality principle
24 (11/20) Martingales and the covergence theorem
25 (11/27) Doob matringales; applications
26 (11/29) Asuma-Hoeffding inequality
27 (12/4)
28 (12/6)
12/17 FINAL EXAM: Monday, December 17 1:30-3:30pm

The literature on Random Processes is vast. In preparing the lectures I use some parts of the following texts, listed in no particular order (you do not need them):

[R] S. M. Ross, Stochastic Processes, New York: J. Wiley & Sons, 1996, and Course Notes by Russel Lyons [Ly]
[V] S. Venkatesh, The Theory of Probability, Cambridge University Press, 2013
[B] P. Billingsley, Probability and Measure, 2nd Ed., New York: J. Wiley & Sons, 1986
[Ro] J. Rosenthal, A First Look at Rigorous Probability Theory, World Scientific, 2006.
[KS] L. Koralov and Y. Sinai, Probability and Random Processes, 2nd Ed., Springer 2007
D. Pollard, A User's Guide to Measure Theoretic Probability, Cambridge University Press, 2002
[S] A. Shiryaev, Probability, 2nd Ed. Springer, 1996
[UI] Class page @ Univ. of Illinois
[L1] and [L2] O. Leveque, Lecture Notes on Probability Part I, Part II