# Random Recursive Equations

WS 2022/2023

## General Information

 Lecture Tuesday 2:00 p.m. - 4:00 p.m. (SRZ 205) Friday 2:00 p.m. -4:00 p.m. (SRZ 205) Exercises: Monday 2:00 p.m. - 4:00 p.m. (SRZ 203) Lecturer: Prof. Dr. Gerold Alsmeyer Assistance: Dr. Rodrigo Bazaes KommVV: Lecture Tutorials Course syllabus: Probabilists are often facing the task to determine the asymptotic behavior of a given sequence of random variables, more precisely, to prove its convergence (in a suitable sense) to a limiting variable X, and to find or at least provide information about the distribution (law) of X. Of course, there is no universal approach to accomplish this task, but in situations where the given stochastic sequence exhibits some kind of recursive structure, expressed in form of a so-called random recursive equation, one is naturally prompted to take advantage of this fact in one way or another. Often, one is led to a distributional equation for the limiting variable X, also called stochastic fixed-point equation of which the law of X constitutes a solution. One of the prominent examples is Lindley's equation that characterizes the actual waiting time of a customer in a stationary one-server queue with i.i.d. interarrival times and i.i.d. service times (G/G/1 queue). The present course aims at an introduction of the theoretical foundations and  some basic tools how to deal with random recursive equation and to gain information about solutions of their stochastic fixed points. This will be done after the discussion of a number of interesting examples from very different areas in probability including queuing theory, branching processes, the analysis of algorithms to nonlinear time series. Learnweb: Please enroll in the Learnweb course for this lecture. Course assessment: There will be an oral exam at the end of the course. Admission to the oral exam is conditional on obtaining at least 40% of the points of the problem sets.