Student Probability Seminar

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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.