GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES

2012 ◽  
Vol 01 (04) ◽  
pp. 1250013 ◽  
Author(s):  
IOANA DUMITRIU ◽  
ELLIOT PAQUETTE
Keyword(s):  
Gaussian Process ◽  
Covariance Matrix ◽  
Law Of Large Numbers ◽  
Test Functions ◽  
Linear Statistics ◽  
Jacobi Ensemble ◽  
Large Numbers ◽  
The Mean ◽  
Shifted Chebyshev Polynomial ◽  
Global Fluctuations

We study the global fluctuations for linear statistics of the form [Formula: see text] as n → ∞, for C1 functions f, and λ1, …, λn being the eigenvalues of a (general) β-Jacobi ensemble. The fluctuation from the mean [Formula: see text] turns out to be given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.

ASYNCHRONOUS UPDATING RESTORES THE LAW OF LARGE NUMBERS IN GLOBALLY COUPLED SYSTEMS

2002 ◽  
Vol 12 (03) ◽  
pp. 663-669 ◽  
Author(s):  
SUDESHNA SINHA
Keyword(s):  
Law Of Large Numbers ◽  
Mean Field ◽  
Coupled Systems ◽  
Large N ◽  
Large Numbers ◽  
Statistical Behavior ◽  
Remarkable Result ◽  
Asynchronous Updating ◽  
The Mean ◽  
The Law

It was observed in earlier studies, that the mean field of globally coupled maps evolving under synchronous updating rules violated the law of large numbers, and this remarkable result generated widespread research interest. In this work we demonstrate that incorporating increasing degrees of asynchronicity in the updating rules rapidly restores the statistical behavior of the mean field. This is clear from the decay of the mean square deviation of the mean field with respect to lattice size N, for varying degrees of asynchronicity, which shows 1/N behavior upto very large N even when the updating is far from fully asynchronous. This is also evidenced through increasing 1/f2 behavior regimes in the power spectrum of the mean field under increasing asynchronicity.

scholarly journals Generalized Nonlinear Least Squares Method for the Calibration of Complex Computer Code Using a Gaussian Process Surrogate

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 985
Author(s):  
Youngsaeng Lee ◽  
Jeong-Soo Park
Keyword(s):  
Gaussian Process ◽  
Least Squares ◽  
Covariance Matrix ◽  
Process Model ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Computer Code ◽  
Test Functions ◽  
Unknown Parameters ◽  
Complex Computer

The approximated nonlinear least squares (ALS) method has been used for the estimation of unknown parameters in the complex computer code which is very time-consuming to execute. The ALS calibrates or tunes the computer code by minimizing the squared difference between real observations and computer output using a surrogate such as a Gaussian process model. When the differences (residuals) are correlated or heteroscedastic, the ALS may result in a distorted code tuning with a large variance of estimation. Another potential drawback of the ALS is that it does not take into account the uncertainty in the approximation of the computer model by a surrogate. To address these problems, we propose a generalized ALS (GALS) by constructing the covariance matrix of residuals. The inverse of the covariance matrix is multiplied to the residuals, and it is minimized with respect to the tuning parameters. In addition, we consider an iterative version for the GALS, which is called as the max-minG algorithm. In this algorithm, the parameters are re-estimated and updated by the maximum likelihood estimation and the GALS, by using both computer and experimental data repeatedly until convergence. Moreover, the iteratively re-weighted ALS method (IRWALS) was considered for a comparison purpose. Five test functions in different conditions are examined for a comparative analysis of the four methods. Based on the test function study, we find that both the bias and variance of estimates obtained from the proposed methods (the GALS and the max-minG) are smaller than those from the ALS and the IRWALS methods. Especially, the max-minG works better than others including the GALS for the relatively complex test functions. Lastly, an application to a nuclear fusion simulator is illustrated and it is shown that the abnormal pattern of residuals in the ALS can be resolved by the proposed methods.

scholarly journals Asymptotic Hitting Time for a Simple Evolutionary Model of Protein Folding

2005 ◽  
Vol 42 (1) ◽  
pp. 39-51 ◽  
Author(s):  
Véronique Ladret
Keyword(s):  
Protein Folding ◽  
Law Of Large Numbers ◽  
Evolutionary Model ◽  
Hitting Time ◽  
Schematic Model ◽  
Zero Temperature ◽  
Native Conformation ◽  
Case Scenario ◽  
Large Numbers ◽  
The Mean

We consider two versions of a simple evolutionary algorithm (EA) model for protein folding at zero temperature, namely the (1 + 1)-EA on the LeadingOnes problem. In this schematic model, the structure of the protein, which is encoded as a bit-string of length n, is evolved to its native conformation through a stochastic pathway of sequential contact bindings. We study the asymptotic behavior of the hitting time, in the mean case scenario, under two different mutations: the one-flip, which flips a unique bit chosen uniformly at random in the bit-string, and the Bernoulli-flip, which flips each bit in the bit-string independently with probability c/n, for some c ∈ ℝ+ (0 ≤ c ≤ n). For each algorithm, we prove a law of large numbers, a central limit theorem, and compare the performance of the two models.

scholarly journals A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages

2019 ◽  
Vol 23 ◽  
pp. 638-661 ◽  
Author(s):  
Aline Marguet
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Law Of Large Numbers ◽  
Empirical Distribution ◽  
Mean Value ◽  
Empirical Measure ◽  
Structured Population ◽  
Large Numbers ◽  
The Mean ◽  
Varying Environment

We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineage. Our approach relies on ergodicity arguments for this time-inhomogeneous Markov process. We apply this technique on the example of a size-structured population with exponential growth in varying environment.

scholarly journals Asymptotic Hitting Time for a Simple Evolutionary Model of Protein Folding

2005 ◽  
Vol 42 (01) ◽  
pp. 39-51 ◽  
Author(s):  
Véronique Ladret
Keyword(s):  
Protein Folding ◽  
Law Of Large Numbers ◽  
Evolutionary Model ◽  
Hitting Time ◽  
Schematic Model ◽  
Zero Temperature ◽  
Native Conformation ◽  
Case Scenario ◽  
Large Numbers ◽  
The Mean

We consider two versions of a simple evolutionary algorithm (EA) model for protein folding at zero temperature, namely the (1 + 1)-EA on the LeadingOnes problem. In this schematic model, the structure of the protein, which is encoded as a bit-string of lengthn, is evolved to its native conformation through a stochastic pathway of sequential contact bindings. We study the asymptotic behavior of the hitting time, in the mean case scenario, under two different mutations: theone-flip, which flips a unique bit chosen uniformly at random in the bit-string, and theBernoulli-flip, which flips each bit in the bit-string independently with probabilityc/n, for somec∈ℝ+(0 ≤c≤n). For each algorithm, we prove a law of large numbers, a central limit theorem, and compare the performance of the two models.

scholarly journals Means as Improper Integrals

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 284
Author(s):  
John Gray ◽  
Andrew Vogt
Keyword(s):  
Law Of Large Numbers ◽  
Cauchy Distribution ◽  
Cumulative Distribution ◽  
Tail Probabilities ◽  
Strong Law ◽  
Improper Integral ◽  
Heavy Tailed Distributions ◽  
Large Numbers ◽  
The Mean ◽  
Heavy Tailed

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

scholarly journals EVALUATION OF THE CONVERGENCE SPEED IN THE LAW OF LARGE NUMBERS FOR GAMMA-DISTRIBUTED SEQUENCES

2018 ◽  
pp. 28-54 ◽  
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Mean Value ◽  
Minimum Volume ◽  
Average Value ◽  
Large Numbers ◽  
The Mean ◽  
The Law ◽  
The Difference

The paper considers the problem of estimating the rate of convergence in the law of large numbers for the case when the initial set of random variables is distributed according to the law of the gamma distribution. The problem is urgent due to the fact that with a small number of initial random variables, accurate and close to the true values are the values obtained on the basis of averaging, in particular, if the receipt of each additional value is associated with significant resource costs. The main result of the paper contains estimates for the modulus of difference in distribution function of the mean value for the set of N random variables in the original population, where N is arbitrary, and distribution function of their limiting value, which is a constant (mean value). The result includes three cases: when the argument of distribution function is greater than the average value; when it is equal to it and when it is less than the average value. Estimates are obtained for the modulus of difference of distributions, which depend not only on the number of random variables N, but also on the argument of distribution function. The dependence of the obtained estimate on the argument of distribution function has an exponential character, and on the volume of the set N this dependence makes about the root of N. For convenience of practical application, and also for solving the inverse problem on the basis of the obtained result, estimating the modulus of the difference of distributions is simplified. On the basis of the simplified estimates obtained, the solution of the following inverse problem is given: to find the minimum volume of the string N at which the modulus of the difference of distributions (the accuracy of estimating the mean value on the basis of the mean value) does not exceed a given (small) value. The paper presents a formula for finding the specified minimum volume N, and an algorithm for finding the exact value of N for the estimate under consideration.

scholarly journals Two probability theorems and their application to some first passage problems

1964 ◽  
Vol 4 (2) ◽  
pp. 214-222 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
First Passage ◽  
Sums Of Random Variables ◽  
Large Numbers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

Modelling of repeatability phenomena using the stochastic ellipsoid approach

Robotica ◽  
2005 ◽  
Vol 24 (4) ◽  
pp. 477-490 ◽  
Author(s):  
Jean-François Brethé ◽  
Eric Vasselin ◽  
Dimitri Lefebvre ◽  
Brayima Dakyo
Keyword(s):  
Gaussian Process ◽  
Covariance Matrix ◽  
Angular Position ◽  
Jump Process ◽  
Second Order ◽  
Stationary Gaussian Process ◽  
Industrial Manipulator ◽  
The Mean

A stochastic ellipsoid modelling of repeatability is proposed for industrial manipulator robots. The covariance matrix of angular position is determined introducing the jump process, which reveals to be a first and second order stationary Gaussian process.From this accurate covariance matrix, the stochastic ellipsoid theory gives the density of position in the workspace around the mean position. Hence the pose repeatability index can be computed in different locations. Computed and experimental repeatability are compared. Experimental repeatability variability is studied. A new “intrinsic repeatability index” is proposed. In conclusion, the modelling reflects well the location and load influence on the repeatability.

scholarly journals Minimal waiting times in static traffic control

2003 ◽  
Vol 7 (1) ◽  
pp. 11-28
Author(s):  
O. Moeschlin ◽  
C. Poppinga
Keyword(s):  
Waiting Time ◽  
Traffic Control ◽  
Law Of Large Numbers ◽  
Waiting Times ◽  
Strong Law ◽  
Poisson Arrival ◽  
Large Numbers ◽  
Arrival Processes ◽  
The Mean ◽  
Traffic Organization

The paper discusses the question of the optimal control of an unsymmetric bottleneck system with Poisson arrival processes having the minimization of the mean individual waiting time as objective. The setup allows the straightforward generalization to more complicated forms of traffic organization. The notion of the mean individual waiting time is based on a theorem of the Little type, which is derived by a strong law of large numbers. The proof makes use of McNeil's formula, which connects the expected total waiting time with the expected queue length.

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