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 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.

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 Strong Law of Large Numbers of the Offspring Empirical Measure for Markov Chains Indexed by Homogeneous Tree

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Huilin Huang
Keyword(s):  
Markov Chains ◽  
Law Of Large Numbers ◽  
Empirical Measure ◽  
Homogeneous Tree ◽  
Almost Everywhere Convergence ◽  
Strong Law ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere ◽  
Large Numbers

We study the limit law of the offspring empirical measure and for Markov chains indexed by homogeneous tree with almost everywhere convergence. Then we prove a Shannon-McMillan theorem with the convergence almost everywhere.

scholarly journals A Note on Rényi's ‘Record’ Problem and Engel's Series

2018 ◽  
Vol 61 (2) ◽  
pp. 363-369 ◽  
Author(s):  
Lulu Fang ◽  
Min Wu
Keyword(s):  
Number Theory ◽  
Large Deviations ◽  
Markov Processes ◽  
Classical Limit ◽  
Limit Theorems ◽  
London Math ◽  
Law Of Large Numbers ◽  
Record Times ◽  
Iterated Logarithm ◽  
Large Numbers

AbstractIn 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

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 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.

The Stochastic and Deterministic Model of the Spread of a Disease in Age-Structured Population

2020 ◽  
Vol 8 (12) ◽  
pp. 290-295
Author(s):  
Yardjouma Yéo
Keyword(s):  
Stochastic Model ◽  
Law Of Large Numbers ◽  
Deterministic Model ◽  
Characteristic Curve ◽  
Structured Population ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Age Structured ◽  
Age Structured Population

We model the spread of an epidemic which depends on age and time. The behavior of the model obtained was the object of our studies.This studies shown that on each characteristic curve, the central limit theorem and the law of large numbers were satisfied. In this paper, we write a stochastic model to studie a disease which is begin in a population which is not large. we study overall the stochastic model obtained by varying the age and time together. We prove that the law of large numbers is satisfied.

A central limit theorem for random coefficient autoregressive models and ARCH/GARCH models

1998 ◽  
Vol 30 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Andreas Rudolph
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Markov Processes ◽  
Law Of Large Numbers ◽  
Random Systems ◽  
Autoregressive Models ◽  
Garch Models ◽  
Conditional Heteroscedasticity ◽  
Large Numbers

In this paper we study the so-called random coeffiecient autoregressive models (RCA models) and (generalized) autoregressive models with conditional heteroscedasticity (ARCH/GARCH models). Both models can be represented as random systems with complete connections. Within this framework we are led (under certain conditions) to CL-regular Markov processes and we will give conditions under which (i) asymptotic stationarity, (ii) a law of large numbers and (iii) a central limit theorem can be shown for the corresponding models.

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