Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain

1968 ◽  
Vol 9 (2) ◽  
pp. 101-111 ◽  
Author(s):  
Mark Pinsky
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Differential Equations ◽  
Small Parameter ◽  
Central Limit ◽  
Finite Markov Chain ◽  
The Central Limit Theorem

VIII.—The Central Limit Theorem for a Convergent Non-homogeneous Finite Markov Chain

1959 ◽  
Vol 65 (2) ◽  
pp. 109-120
Author(s):  
J. L. Mott
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Stochastic Matrix ◽  
Homogeneous Markov Chain ◽  
Finite Markov Chain ◽  
The Central Limit Theorem

SynopsisThe distribution of xn, the number of occurrences of a given one of k possible states of a non-homogeneous Markov chain {Pj} in n successive trials, is considered. It is shown that if Pn → P, a positive-regular stochastic matrix, as n → ∞ then the distribution about its mean of xn/n½ tends to normality, and that the variance tends to that of the corresponding distribution associated with the homogeneous chain {P}.

scholarly journals Central limit theorem for a class of SPDEs

2015 ◽  
Vol 52 (3) ◽  
pp. 786-796 ◽  
Author(s):  
Parisa Fatheddin
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Partial Differential Equations ◽  
Limit Theorem ◽  
Differential Equations ◽  
Central Limit ◽  
Stochastic Partial Differential Equations ◽  
Population Models ◽  
The Central Limit Theorem ◽  
Super Brownian Motion

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.

scholarly journals Central limit theorem for a class of SPDEs

2015 ◽  
Vol 52 (03) ◽  
pp. 786-796 ◽  
Author(s):  
Parisa Fatheddin
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Partial Differential Equations ◽  
Limit Theorem ◽  
Differential Equations ◽  
Central Limit ◽  
Stochastic Partial Differential Equations ◽  
Population Models ◽  
The Central Limit Theorem ◽  
Super Brownian Motion

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.

Examples and Counterexamples

2019 ◽  
pp. 448-464
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Type Inequality ◽  
Stationary Sequence ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
The Moment

In this chapter we further comment on the sharpness of several results presented in this monograph, by presenting examples and counterexamples. We study first the moment properties of the renewal Markov chain introduced in Chapter 11. This allows us to show that the Maxwell–Woodroofe projective condition introduced in Chapter 4 is essentially optimal for the partial sums of a stationary sequence in L2 to satisfy the central limit theorem under the standard normalization √n. Moreover, we also investigate the sharpness of the Burkholder-type inequality developed in Chapter 3 via Maxwell–Woodroofe-type characteristics. In the last part of this chapter, we analyze several telescopic-type examples allowing us to elucidate the fact that a CLT behavior does not imply its functional form under any normalization. Even in the case when the variance of the partial sums is linear in n, the CLT does not necessarily imply the invariance principle.

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