Class number 054–39482
Class meetings: MW, 9:30am-12:20pm
in KAP 138
Information on this and related pages changes frequently.
Instructor: Sergey Lototsky.
Office: KAP 248D.
Phone: 213 740 2389
E-mail: lototsky usc edu.
Office Hours: MW after the class. Appointments at other time are welcome.
Course objective: to survey theory and applications of random processes that are invariant in distribution
under linear scaling of time and space.
More general goal: to learn something interesting, new, and/or useful.
Official grading scheme: 20% in-class work, 40% homework assignments, 40% final presentation.
Main references:
- Paul Embrechts and Makoto Maejima. Selfsimilar processes. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2002. xii+111 pp. ISBN: 0-691-09627-9
- Ciprian Tudor. Non-Gaussian selfsimilar stochastic processes. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham, [2023], xii+101 pp. ISBN: 978-3-031-33771-0; 978-3-031-33772-7. This book is available in electronic form from the USC Libraries
Other references:
- Søren Asmussen and Peter W. Glynn. Stochastic simulation: algorithms and analysis. Stochastic Modelling and Applied Probability, 57. Springer, New York, 2007. xiv+476 pp. ISBN: 978-0-387-30679-7
- Jan Beran. Statistics for long-memory processes. Monographs on Statistics and Applied Probability, 61. Chapman and Hall, New York, 1994. x+315 pp. ISBN: 0-412-04901-5
An example of a book review from Math reviews [Edition 1, Edition 2] and from the Bulletin of the AMS
The class notes
Supplementary notes: mine
- General summary of probability
- Random variables: general definitions and an easy diagram
- Basic inequalities
- More on probability inequalities
- Cauchy’s functional equation
- Stable distributions
- Stochastic analysis in continuous time
- Weak convergence of probability measures
- A summary of characteristic functions
- A summary of local time
- Gamma and Beta functions
- Exact relations among probability distributions
- Gaussian objects: Normal random variables, CLT, and more
- Gaussian distribution: a time line
- A summary of random object generation
- A summary of Gaussian inequalities
- A summary of Brownian motion
- A summary of SODEs
- Orthogonal polynomials
- Two computations: the Basel problem and the Gaussian integral
Supplementary notes: Found on line
- The Garsia-Rodemich-Rumsey Lemma and MR review
- How to generate symmetric alpha-stable random variables [a research paper]
- From one fBM to another
- Fractional Brownian motion: a survey
- A short comprehensive summary of fBM
- Fractional business can be even more complicated
From ChatGPT
- Linear Fractional Stable Motion (not a semi-martingale)
- Log Fractional Stable Motion (and some more)
Our Progress
May 20: Course overview
May 25: Memorial Day (no class)
May 27: Foundations (stochastic continuity, measurable modification, Kolmogorov’s criterion); a definition of self-similarity; examples and counter examples
June 1: Hermite processes (characterization and convergence)
June 3: Chaos expansion (Wiener-Ito vs Cameron-Martin)
June 8: Symmetric alpha-stable H-self-similar processes with stationary increments [or SaS H-sssi, for short] (construction and sample path properties)
June 10: SaS H-sssi (completing the phase diagram and Chapter 3 of the book)