Student Probability Seminar

Conditioning I.I.D. Random Variables to Significantly Exceed Their Mean

Speaker: Matan Harel

Location: Warren Weaver Hall 905

Date: Tuesday, November 12, 2013, 3:30 p.m.

Synopsis:

What happens to the empirical distribution of I.I.D. random variables if we condition on their sum exceeding their mean by a multiplicative constant strictly large than 1? Although the law of large numbers guarantees that such an event will have vanishing probability, the $$S_n > (1 + \delta) E[X_1]$$ has positive probability for any finite $$n$$, where $$S_n$$ is the normalized sum of $$X_i.$$ We will describe the effect of such conditioning for any $$X_i$$ that has squashed or stretched exponential tails - i.e. $$P[X_n >t] ~ e^{-t^\alpha}$$ for some positive $$\alpha$$ and $$t$$ sufficiently large. Specifically, if $$\alpha$$ is greater or equal to 1, we will see a nonvanishing Radon-Nykodyn derivative, known as a Gibbs factor. If $$\alpha$$ is less than 1, however, the effect of conditioning will be the existence of a very large maximum.