scholarly journals Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

2014 ◽  
Vol 03 (03) ◽  
pp. 1450012 ◽  
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.

scholarly journals Large deviations in non-uniformly hyperbolic dynamical systems

2008 ◽  
Vol 28 (2) ◽  
pp. 587-612 ◽  
Author(s):  
LUC REY-BELLET ◽  
LAI-SANG YOUNG
Keyword(s):  
Dynamical Systems ◽  
Generating Functions ◽  
Limit Theorems ◽  
Large Deviation ◽  
Return Times ◽  
Hyperbolic Diffeomorphisms ◽  
Large Deviation Principles ◽  
Hyperbolic Dynamical Systems ◽  
Rank One ◽  
Dispersing Billiards

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.

scholarly journals Limit theorems associated with the Pitman–Yor process

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.

scholarly journals Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

2015 ◽  
pp. 406-418
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorems ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Convergence In Distribution ◽  
Initial State ◽  
Uniform Limit ◽  
Limit Measure ◽  
Long Time ◽  
Discrete Uniform

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

scholarly journals Large deviations for a class of tempered subordinators and their inverse processes

2021 ◽  
pp. 1-21
Author(s):  
Nikolai Leonenko ◽  
Claudio Macci ◽  
Barbara Pacchiarotti
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
Large Deviation ◽  
Common Law ◽  
Random Variables ◽  
Tempered Distributions ◽  
One Dimensional ◽  
Large Deviation Principles ◽  
Time Changes ◽  
Non Gaussian

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper is a non-central moderate deviations result. More precisely we mean a class of large deviation principles that fill the gap between the (trivial) weak convergence of some non-Gaussian identically distributed random variables to their common law, and the convergence of some other related random variables to a constant. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes.

LARGE DEVIATIONS FOR THE LONGEST GAP IN POISSON PROCESSES

2019 ◽  
Vol 101 (1) ◽  
pp. 146-156 ◽  
Author(s):  
JOSEPH OKELLO OMWONYLEE
Keyword(s):  
Large Deviations ◽  
Laplace Transform ◽  
Large Deviation ◽  
Poisson Processes ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Large Deviation Principles ◽  
The Laplace Transform ◽  
Rate Functions ◽  
Maximal Time

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

scholarly journals An Explicit Large Deviation Analysis of the Spatial Cycle Huang–Yang–Luttinger Model

2021 ◽  
Vol 22 (5) ◽  
pp. 1535-1560
Author(s):  
Stefan Adams ◽  
Matthew Dickson
Keyword(s):  
Large Deviation ◽  
Einstein Condensation ◽  
Counter Term ◽  
Luttinger Model ◽  
Deviation Analysis ◽  
Large Deviation Principles ◽  
Reference Process ◽  
Rate Functions ◽  
Explicit Bounds ◽  
Bose Einstein

AbstractWe introduce a family of ‘spatial’ random cycle Huang–Yang–Luttinger (HYL)-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. Here, spatial refers to the Poisson reference process of random cycle weights. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose–Einstein condensation. The main focus is a large deviation analysis for the diverging counter term where we identify three different regimes depending on the scale of divergence with respect to the main large deviation scale. Our analysis derives explicit bounds in critical regimes using the Poisson nature of the random cycle distributions.

JOINT WEIGHTED LIMIT THEOREMS FOR GENERAL DIRICHLET SERIES

2011 ◽  
Vol 16 (1) ◽  
pp. 39-51
Author(s):  
Jonas Genys ◽  
Antanas Laurinčikas
Keyword(s):  
Weak Convergence ◽  
Dirichlet Series ◽  
Limit Theorems ◽  
Random Element ◽  
Probability Measures ◽  
Limit Measure ◽  
Additional Hypothesis ◽  
General Dirichlet Series ◽  
Convergence Of Probability Measures ◽  
Weighted Limit

In the paper,two joint weighted limit theorems in the sense of weak convergence of probability measures on the complex plane for general Dirichlet series are obtained. The first of them gives only the existence of the limit measure, while in the second theorem,under some additional hypothesis on the weight function, the explicit form of the limit measure is presented. Namely, the limit measure coincides with the distribution of some random element related to considered Dirichlet series.

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