Conditional law of large numbers and entropy of Markov processes

2020 ◽  
Vol 23 (02) ◽  
pp. 2050011
Author(s):  
Jinwen Chen ◽  
Sen Dan
Keyword(s):  
Compact Operator ◽  
Markov Processes ◽  
Law Of Large Numbers ◽  
Large Numbers ◽  
Operator Argument

The importance of a law of large numbers is well known. In this paper, conditional law of large numbers for Markov processes is proved, which can be used in computing quantities related to sub-Markov sequences. A variational interpretation for this limit is given, which shows that typically the quantities of interests tend to minimize a certain entropy. A quasi-compact operator argument is involved.

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 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.

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.

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

1998 ◽  
Vol 30 (01) ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document