An application of chain-dependent processes to meteorology

1977 ◽  
Vol 14 (3) ◽  
pp. 598-603 ◽  
Author(s):  
Richard W. Katz
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Explicit Formula ◽  
Probabilistic Model ◽  
Markov Chain Model ◽  
Chain Model ◽  
The Central Limit Theorem ◽  
Normalizing Constant ◽  
Dependent Processes

An explicit formula is derived for the variance normalizing constant in the central limit theorem for chain-dependent processes. As an application to meteorology, a specific chain-dependent process is proposed as a probabilistic model for the sequence of daily amounts of precipitation. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation.

An application of chain-dependent processes to meteorology

1977 ◽  
Vol 14 (03) ◽  
pp. 598-603 ◽  
Author(s):  
Richard W. Katz
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Explicit Formula ◽  
Probabilistic Model ◽  
Markov Chain Model ◽  
Chain Model ◽  
The Central Limit Theorem ◽  
Normalizing Constant ◽  
Dependent Processes

An explicit formula is derived for the variance normalizing constant in the central limit theorem for chain-dependent processes. As an application to meteorology, a specific chain-dependent process is proposed as a probabilistic model for the sequence of daily amounts of precipitation. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation.

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}.

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.

CLTs for Dependent Processes

2021 ◽  
pp. 548-567
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Strong Mixing ◽  
Mixing Processes ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Dependent Processes

This chapter deals with the central limit theorem (CLT) for dependent processes. As with the law of large numbers, the focus is on near‐epoch dependent and mixing processes and array versions of the results are given to allow heterogeneity. The cornerstone of these results is a general CLT due to McLeish, from which a result for martingales is obtained directly. A result for stationary ergodic mixingales is given, and the rest of the chapter is devoted to proving and interpreting a CLT for mixingales and hence for arrays that are near‐epoch dependent on a strong‐mixing and uniform-mixing processes.

An Efficient Markov Chain Model for the Simulation of Heterogeneous Soil Structure

2004 ◽  
Vol 68 (2) ◽  
pp. 346 ◽  
Author(s):  
Keijan Wu ◽  
Naoise Nunan ◽  
John W. Crawford ◽  
Iain M. Young ◽  
Karl Ritz
Keyword(s):  
Markov Chain ◽  
Soil Structure ◽  
Markov Chain Model ◽  
Chain Model ◽  
Heterogeneous Soil
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