Florian Angeletti

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F. Angeletti, H. Touchette, E. Bertin, P. Abry

Large deviations for correlated random variables described by a matrix product ansatz,

F. Angeletti, H. Touchette, E. Bertin, P. Abry, JSTAT, (2014), 2003,
doi:10.1088/1742-5468/2014/02/P02003, arxiv:1310.6952, BIB, PDF

Abstract

We study the large deviations of sums of correlated random variables described by a matrix product ansatz,which generalizes the product structure of independent random variables to matrices whose non-commutativity is the source of correlations. We show with specific examples that different large deviation behaviors can be found with this ansatz. In particular,it is possible to construct sums of correlated random variables that violate the Law of Large Numbers,the Central Limit Theorem,as well as sums that have nonconvex rate functions or rate functions with linear parts or plateaux.

F. Angeletti, E. Bertin, P. Abry

General limit distributions for sums of random variables with a matrix product representation,

F. Angeletti, E. Bertin, P. Abry, Journal of Statistical Physics, (2014), 1255-1283,
doi:10.1007/s10955-014-1111-y, arxiv:1406.5016, BIB, PDF

Abstract

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism,non-standard limit distributions are obtained,and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions.