On Limit Theorems for Random Fuzzy Sets Including Large Deviation Principles

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

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Limit Theorems for Sums of Random Fuzzy Sets

Journal of Mathematical Analysis and Applications ◽

10.1006/jmaa.2000.7428 ◽

2001 ◽

Vol 259 (2) ◽

pp. 554-565 ◽

Cited By ~ 18

Author(s):

Marco Dozzi ◽

Ely Merzbach ◽

Volker Schmidt

Keyword(s):

Fuzzy Sets ◽

Limit Theorems ◽

Random Fuzzy Sets

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Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

Random Matrices Theory and Application ◽

10.1142/s2010326314500129 ◽

2014 ◽

Vol 03 (03) ◽

pp. 1450012 ◽

Cited By ~ 1

Author(s):

Jan Nagel

Keyword(s):

Weak Convergence ◽

Spectral Measure ◽

Limit Theorems ◽

Large Deviation ◽

The Other ◽

Limit Measure ◽

Jacobi Ensemble ◽

Large Deviation Principles ◽

Beta Ensemble ◽

Rate Functions

In this paper, we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension n. In these cases, the limit measure is given by the Marchenko–Pastur law and the semicircle law, respectively. For the weighted spectral measure, we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.

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Limit theorems associated with the Pitman–Yor process

Advances in Applied Probability ◽

10.1017/apr.2017.13 ◽

2017 ◽

Vol 49 (2) ◽

pp. 581-602

Author(s):

Shui Feng ◽

Fuqing Gao ◽

Youzhou Zhou

Keyword(s):

Disordered Systems ◽

Limit Theorems ◽

Large Deviation ◽

Dirichlet Distribution ◽

Discrete Measure ◽

Random Energy Model ◽

Large Deviation Principles ◽

Large Numbers ◽

Random Weights ◽

Two Parameter

Abstract The Pitman–Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson–Dirichlet distribution with parameters 0 < α < 1, θ > -α. The parameters α and θ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability η. In this paper we consider the limit theorems for the Pitman–Yor process and the two-parameter Poisson–Dirichlet distribution. These include the law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either α tending to 0 or 1. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to the generalized random energy model for disordered systems.

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