scholarly journals Large Deviation Theorem for Branches of the Random Binary Tree in the Horton--Strahler Analysis

2020 ◽  
Vol 34 (1) ◽  
pp. 938-949
Author(s):  
Ken Yamamoto
Keyword(s):  
Binary Tree ◽  
Large Deviation ◽  
Large Deviation Theorem

On the limiting distribution of the failure time of fibrous materials

1983 ◽  
Vol 15 (02) ◽  
pp. 331-348
Author(s):  
Wagner De Souza Borges
Keyword(s):  
Materials Science ◽  
Failure Time ◽  
Large Deviation ◽  
Limiting Distribution ◽  
Statistical Characteristics ◽  
Fibrous Materials ◽  
Large Deviation Theorem ◽  
Science Area ◽  
The Individual ◽  
Failure Time Distributions

A large deviation theorem of the Cramér–Petrov type and a ranking limit theorem of Loève are used to derive an approximation for the statisticaldistribution of the failure time of fibrous materials. For that, fibrousmaterials are modeled as a series of independent and identical bundles of parallel filaments and the asymptotic distribution of their failure time is determined in terms of statistical characteristics of the individual filaments, as both the number of filaments in each bundle and the number of bundles in the chain grow large simultaneously. While keeping the numbernof filaments in each bundle fixed and increasing only the chain lengthkleads to a Weibull limiting distribution for the failure time, letting both increase in such a way that logk(n)= o(n), we show that the limit distribution isfor. Since fibrous materials which are both long and have many filaments prevail, the result is of importance in the materials science area since refined approximations to failure-time distributions can be achieved.

On the limiting distribution of the failure time of fibrous materials

1983 ◽  
Vol 15 (2) ◽  
pp. 331-348 ◽  
Author(s):  
Wagner De Souza Borges
Keyword(s):  
Materials Science ◽  
Failure Time ◽  
Large Deviation ◽  
Limiting Distribution ◽  
Statistical Characteristics ◽  
Fibrous Materials ◽  
Large Deviation Theorem ◽  
Science Area ◽  
The Individual ◽  
Failure Time Distributions

A large deviation theorem of the Cramér–Petrov type and a ranking limit theorem of Loève are used to derive an approximation for the statistical distribution of the failure time of fibrous materials. For that, fibrous materials are modeled as a series of independent and identical bundles of parallel filaments and the asymptotic distribution of their failure time is determined in terms of statistical characteristics of the individual filaments, as both the number of filaments in each bundle and the number of bundles in the chain grow large simultaneously. While keeping the number n of filaments in each bundle fixed and increasing only the chain length k leads to a Weibull limiting distribution for the failure time, letting both increase in such a way that log k(n) = o(n), we show that the limit distribution is for . Since fibrous materials which are both long and have many filaments prevail, the result is of importance in the materials science area since refined approximations to failure-time distributions can be achieved.

Moderate- and large-deviation probabilities in actuarial risk theory

1989 ◽  
Vol 21 (04) ◽  
pp. 725-741 ◽  
Author(s):  
Eric Slud ◽  
Craig Hoesman
Keyword(s):  
Strong Approximation ◽  
Large Deviation ◽  
Age Distribution ◽  
Risk Theory ◽  
Renewal Processes ◽  
Collective Risk ◽  
Large Deviation Theorem ◽  
Large Deviation Probabilities ◽  
Actuarial Risk ◽  
Portfolio Size

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

Large deviations in Axiom A endomorphisms

2003 ◽  
Vol 133 (6) ◽  
pp. 1379-1388 ◽  
Author(s):  
Pei-Dong Liu ◽  
Min Qian ◽  
Yang Zhao
Keyword(s):  
Large Deviations ◽  
Large Deviation ◽  
Axiom A ◽  
Space Technique ◽  
Large Deviation Theorem ◽  
Smale Space

By applying a general large-deviation theorem of Kifer and Ruelle's Smale space technique, some large-deviation estimates are proved for Axiom A endomorphisms.

Moderate- and large-deviation probabilities in actuarial risk theory

1989 ◽  
Vol 21 (4) ◽  
pp. 725-741 ◽  
Author(s):  
Eric Slud ◽  
Craig Hoesman
Keyword(s):  
Strong Approximation ◽  
Large Deviation ◽  
Age Distribution ◽  
Risk Theory ◽  
Renewal Processes ◽  
Collective Risk ◽  
Large Deviation Theorem ◽  
Large Deviation Probabilities ◽  
Actuarial Risk ◽  
Portfolio Size

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

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.

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