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.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (1) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

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 Law of large numbers for the sequence of uncorrelated fuzzy random variables

2021 ◽  
pp. 1-8
Author(s):  
Li Guan ◽  
Jinping Zhang ◽  
Jieming Zhou
Keyword(s):  
Law Of Large Numbers ◽  
Hausdorff Metric ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Strong Law ◽  
Fuzzy Variables ◽  
Large Numbers ◽  
The Law ◽  
Fuzzy Random

This work proposes the concept of uncorrelation for fuzzy random variables, which is weaker than independence. For the sequence of uncorrelated fuzzy variables, weak and strong law of large numbers are studied under the uniform Hausdorff metric d H ∞ . The results generalize the law of large numbers for independent fuzzy random variables.

scholarly journals The law of large numbers for free identically distributed random variables

1996 ◽  
Vol 24 (1) ◽  
pp. 453-465 ◽  
Author(s):  
Hari Bercovici ◽  
Vittorino Pata
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Large Numbers ◽  
The Law

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 On the rate of convergence in the strong law of large numbers for arrays

1992 ◽  
Vol 45 (3) ◽  
pp. 479-482 ◽  
Author(s):  
Tien-Chung Hu ◽  
N.C. Weber
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Second Moment ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

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