scholarly journals Large deviation rate functions for the partition function in a log-gamma distributed random potential

2013 ◽  
Vol 41 (6) ◽  
pp. 4248-4286 ◽  
Author(s):  
Nicos Georgiou ◽  
Timo Seppäläinen
Keyword(s):  
Partition Function ◽  
Large Deviation ◽  
Random Potential ◽  
Deviation Rate ◽  
Rate Functions

scholarly journals Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

2019 ◽  
Vol 26 (3) ◽  
pp. 195-209 ◽  
Author(s):  
Frank Kwasniok
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Finite Time ◽  
Large Deviation ◽  
Atmospheric Model ◽  
Lyapunov Spectrum ◽  
Finite Time Lyapunov Exponents ◽  
Fluctuation Field ◽  
Deviation Rate ◽  
Rate Functions

Abstract. The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

scholarly journals Large-Deviation Results for Discriminant Statistics of Gaussian Locally Stationary Processes

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Junichi Hirukawa
Keyword(s):  
Likelihood Ratio ◽  
Quadratic Forms ◽  
Large Deviation ◽  
Stationary Processes ◽  
Deviation Rate ◽  
Log Likelihood ◽  
Log Likelihood Ratio ◽  
Rate Functions ◽  
Locally Stationary Processes ◽  
Mean Function

This paper discusses the large-deviation principle of discriminant statistics for Gaussian locally stationary processes. First, large-deviation theorems for quadratic forms and the log-likelihood ratio for a Gaussian locally stationary process with a mean function are proved. Their asymptotics are described by the large deviation rate functions. Second, we consider the situations where processes are misspecified to be stationary. In these misspecified cases, we formally make the log-likelihood ratio discriminant statistics and derive the large deviation theorems of them. Since they are complicated, they are evaluated and illustrated by numerical examples. We realize the misspecification of the process to be stationary seriously affecting our discrimination.

scholarly journals The height of the Lyndon tree

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lucas Mercier ◽  
Philippe Chassaing
Keyword(s):  
Binary Tree ◽  
Large Deviation ◽  
Random Variables ◽  
Convergence In Probability ◽  
Eulerian Numbers ◽  
Deviation Rate ◽  
Rate Functions ◽  
International Audience ◽  
Lyndon Words

International audience We consider the set $\mathcal{L}_n<$ of n-letters long Lyndon words on the alphabet $\mathcal{A}=\{0,1\}$. For a random uniform element ${L_n}$ of the set $\mathcal{L}_n$, the binary tree $\mathfrak{L} (L_n)$ obtained by successive standard factorization of $L_n$ and of the factors produced by these factorization is the $\textit{Lyndon tree}$ of $L_n$. We prove that the height $H_n$ of $\mathfrak{L} (L_n)$ satisfies $\lim \limits_n \frac{H_n}{\mathsf{ln}n}=\Delta$, in which the constant $\Delta$ is solution of an equation involving large deviation rate functions related to the asymptotics of Eulerian numbers ($\Delta ≃5.092\dots $). The convergence is the convergence in probability of random variables.

scholarly journals Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

2018 ◽  
Author(s):  
Frank Kwasniok
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Finite Time ◽  
Large Deviation ◽  
Atmospheric Model ◽  
Integration Time ◽  
Legendre Transform ◽  
Finite Time Lyapunov Exponents ◽  
Deviation Rate ◽  
Rate Functions

Abstract. The stability properties as characterised by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. An empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The two leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are independent of the integration time of the finite-time Lyapunov exponents. Then large-deviation rate functions are estimated from time series of daily Lyapunov exponents using the Legendre transform and from time series of Lyapunov exponents with long integration times based on their probability density function. Serial correlation in the time series is properly accounted for. Convergence to a large-deviation principle can be established for all of the Lyapunov exponents which is rather slow for the most unstable exponents and becomes faster when going further down in the Lyapunov spectrum. Convergence is generally faster for the Gaussian behaviour in the vicinity of the mean value. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also joint large-deviation rate functions beyond the Gaussian approximation are calculated for the first and the second Lyapunov exponent.

scholarly journals Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

2015 ◽  
Vol 52 (1) ◽  
pp. 68-81 ◽  
Author(s):  
K. M. Kosiński ◽  
M. Mandjes
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Random Vectors ◽  
Deviation Principle ◽  
Logarithmic Asymptotics ◽  
Rate Functions ◽  
Regularly Varying ◽  
Positive Index

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

scholarly journals Dynamical Phase Transitions for Flows on Finite Graphs

2020 ◽  
Vol 181 (6) ◽  
pp. 2353-2371
Author(s):  
Davide Gabrielli ◽  
D. R. Michiel Renger
Keyword(s):  
Large Deviation ◽  
Time Window ◽  
Scaling Limit ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Dynamical Phase Transition ◽  
Dynamical Phase ◽  
Hydrodynamic Scaling ◽  
Deviation Rate ◽  
Dynamical Phase Transitions

AbstractWe study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.

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