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

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 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 Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

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