## Wednesday 4 May 2011 - Markus Fischer

**ARGOMENTI:** Seminars Ph.D. Program

SEMINARIO DOTTORATO

Wednesday 4 May 2011 h. 14:30, room 2BC30

Markus Fischer (Dip. Mat.)

"Large Deviations in Probability Theory"

-Abstract

In probability theory, the term large deviations refers to an asymptotic property of the laws of families of random variables depending on a large deviations parameter.

A classical example is derived from coin flipping. For each number n, consider the random experiment of tossing n coins. Let S(n) denote the number of coins that land heads up. The quantities S(n) and S(n)/n are random variables, S(n)/n being the empirical mean, here equal to the empirical probability of getting heads. If the coins are fair and tossed independently, then by the law of large numbers S(n)/n will converge to 1/2 as n tends to infinity. Consequently, given any strictly positive c, the probability that S(n)/n is greater than 1/2+c (or less than 1/2-c) goes to zero as n tends to infinity. But one can say more about the convergence of those probabilities of deviation from the law of large numbers limit. Indeed, the decay to zero is exponentially fast (in the large deviations parameter n) with rates that can be determined exactly. The exponential decay of deviation probabilities is a common property of families of random objects arising in many different contexts.

The aim of this talk is to give an introduction to the theory of large deviations, illustrating it by elementary examples as well as in the context of a class of mean field models.

Rif. int. C. Marastoni, T. Vargiolu

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