On the concentration of large deviations for fat tailed distributions

Details On the concentration of large deviations for fat tailed distributions On the concentration of large deviations for fat tailed distributions

2012-01-13

arxiv.org/abs/1201.2817v1

Physics

Condensed Matter

Statistical Mechanics

8 pagss, 1 figure

Scientific paper

For fat tailed distributions (i.e. those that decay slower than an exponential), large deviations not only become relatively likely, but the way in which they are realized changes dramatically: A finite fraction of the whole sample deviation is concentrated on a single variable: large deviations are not the accumulation of many small deviations, but rather they are dominated to a single large fluctuation. The regime of large deviations is separated from the regime of typical fluctuations by a phase transition where the symmetry between the points in the sample is {\em spontaneously broken}. This phenomenon has been discussed in the context of mass transport models in physics, where it takes the form of a condensation phase transition. Yet, the phenomenon is way more general. For example, in risk management of large portfolios, it suggests that one should expect losses to concentrate on a single asset: when extremely bad things happen, it is likely that there is a single factor on which bad luck concentrates. Along similar lines, one should expect that bubbles in financial markets do not gradually deflate, but rather burst abruptly and that in the most rainy day of a year, precipitation concentrate on a given spot. Analogously, when applied to biological evolution, we're lead to infer that, if fitness changes for individual mutations have a broad distribution, those large deviations that lead to better fit species are not likely to result from the accumulation of small positive mutations. Rather they are likely to arise from large rare jumps.

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