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)

**TA**s:

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)

Lect. # | Topics | Textbooks | Reading | HW | Solutions | |
---|---|---|---|---|---|---|

1 (8/28) | Course description. What is probability, by example: Borel's Normal Numbers |
[G], Sec.1.1, 1.8 | [H] 11.1, 11.2Billingsley [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 | HW1 | Solutions | |

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.1.8 | HW2 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 |
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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) |
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23 (11/15) | Conditional expectation Filtrations; orthogonality principle |
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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