On controlled one-dimensional diffusion processes with unknown parameter

1977 ◽  
Vol 9 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Věra Dufková
Keyword(s):  
Unknown Parameter ◽  
Diffusion Processes ◽  
Asymptotic Properties ◽  
Control Policy ◽  
Controlled Diffusion ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Stationary Control ◽  
Optimal Stationary Control

We consider a controlled diffusion process, the description of which depends on an unknown parameter α, and investigate the following control policy. To each α an optimal stationary control is associated. α is estimated recurrently from the trajectory by Bayes' method, and the optimal stationary control corresponding to the estimate is used. We establish the consistency of the estimate, and present asymptotic properties of the criterion function. They follow from the central limit theorem, from the law of large numbers and from the law of the iterated logarithm for local martingales.

On controlled one-dimensional diffusion processes with unknown parameter

1977 ◽  
Vol 9 (01) ◽  
pp. 105-124 ◽  
Author(s):  
Věra Dufková
Keyword(s):  
Unknown Parameter ◽  
Diffusion Processes ◽  
Asymptotic Properties ◽  
Control Policy ◽  
Controlled Diffusion ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Stationary Control ◽  
Optimal Stationary Control

We consider a controlled diffusion process, the description of which depends on an unknown parameter α, and investigate the following control policy. To each α an optimal stationary control is associated. α is estimated recurrently from the trajectory by Bayes' method, and the optimal stationary control corresponding to the estimate is used. We establish the consistency of the estimate, and present asymptotic properties of the criterion function. They follow from the central limit theorem, from the law of large numbers and from the law of the iterated logarithm for local martingales.

Estimation and control in Markov chains

1974 ◽  
Vol 6 (1) ◽  
pp. 40-60 ◽  
Author(s):  
P. Mandl
Keyword(s):  
Markov Chains ◽  
Asymptotic Properties ◽  
Control Policy ◽  
Contrast Method ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Estimation And Control ◽  
And Control ◽  
Stationary Control ◽  
Optimal Stationary Control

We consider a finite controlled Markov chain, the description of which depends on an unknown parameter a, and investigate the following control policy. To each a an optimal stationary control is associated. a is estimated recurrently from the trajectory by the minimum contrast method, and the optimal stationary control corresponding to the estimate is used. We present asymptotic properties of the estimate and of the criterion function. They follow from the law of large numbers and from the central limit theorem for controlled Markov chains derived with the aid of martingales.

Estimation and control in Markov chains

1974 ◽  
Vol 6 (01) ◽  
pp. 40-60 ◽  
Author(s):  
P. Mandl
Keyword(s):  
Markov Chains ◽  
Asymptotic Properties ◽  
Control Policy ◽  
Contrast Method ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Estimation And Control ◽  
And Control ◽  
Stationary Control ◽  
Optimal Stationary Control

We consider a finite controlled Markov chain, the description of which depends on an unknown parameter a, and investigate the following control policy. To each a an optimal stationary control is associated. a is estimated recurrently from the trajectory by the minimum contrast method, and the optimal stationary control corresponding to the estimate is used. We present asymptotic properties of the estimate and of the criterion function. They follow from the law of large numbers and from the central limit theorem for controlled Markov chains derived with the aid of martingales.

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.

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.

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