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On the concentration of large deviations for fat tailed distributions, with application to financial data

2014; Institute of Physics; Volume: 2014; Issue: 9 Linguagem: Inglês

10.1088/1742-5468/2014/09/p09030

1742-5468

Mario Filiasi, Giacomo Livan, Matteo Marsili, Maria Peressi, Erik Vesselli, Elia Zarinelli,

Random Matrices and Applications

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. 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 spontaneously broken. For stochastic processes with a fat tailed microscopic noise, this implies that, while typical realizations are well described by a diffusion process with continuous sample paths, large deviation paths are typically discontinuous. For eigenvalues of random matrices with fat tailed distributed elements, a large deviation where the trace of the matrix is anomalously large concentrates on just a single eigenvalue, whereas in the thin tailed world the large deviation affects the whole distribution.