Jordan Stoyanov (Bulgarian Academy of Sciences and Shandong University, China)
24 June 2022 2:00 PM
Herschel Building, Level 4, Teaching Room 2
Characterizations of Probability Distributions: Role of Counterexamples
Speaker: Jordan Stoyanov (Bulgarian Academy of Sciences and Shandong University, China)
Abstract: I will present two amazing stories, (1) and (2), which illustrate well some of the most important ideas and techniques in Modern Probability and Statistics. Fundamental are any kind of Limit theorems. This is a deep beautiful Mathematics with great implications for applications. It is well known that a progress in areas such as Statistical Physics, Random Graphs, Number Theory and Random Matrices was made only by involving the power of the Limit Theorems from Probability Theory.
(1) The simple distributional equation, X + Y = XY, where X and Y are random variables (r.v.s) with unknown distributions, leads to unpredictable and nontrivial answers, which are different in the continuous case and the discrete case.
(2) If we have a sequence of independent and bounded r.v.s, intuitively one can expect that the CLT (central limit theorem) holds, i.e., the centred and normalized sums converge to a r.v. Z ~ N(0,1). We know, the CLT is one of the greatest laws in both, Theory and Nature. You will see a specific stochastic model, related to Number Theory, in which a nonconventional limit theorem holds: the centred and normalized sums converge to a bounded (!) r.v., Z*, whose range of values is the interval [- 3, + 3]. Hence, no CLT. Looks strange, and it is, you will see why.
The next is to discuss on the great role played by “Counterexamples” in the whole Mathematics, in particular, in Probability and Statistics. Specific eye opening and instructive counterexamples will be given in detail. It is not just a satisfaction of our curiosity to know counterexamples. They help us to clearly mark the borders up to which a result is true. You cross the border, you fall into a “trap”.
Several related comments will be made during the talk. A couple of challenging open problems will be outlined.