scholarly journals The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations

2020 ◽  
pp. 1-23
Author(s):  
Dawid Czapla ◽  
Sander C. Hille ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Piecewise Deterministic Markov Process ◽  
Iterated Logarithm ◽  
The Law

scholarly journals The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joanna Kubieniec
Keyword(s):  
Dynamical System ◽  
Markov Chain ◽  
Law Of Iterated Logarithm ◽  
Random Dynamical System ◽  
Piecewise Deterministic Markov Process ◽  
Exponential Ergodicity ◽  
Iterated Logarithm ◽  
State Dependent ◽  
Jump Intensity ◽  
The Law

Abstract In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a state-dependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.

scholarly journals The Law of the Iterated Logarithm for a Markov Process

1970 ◽  
Vol 41 (3) ◽  
pp. 945-955 ◽  
Author(s):  
R. P. Pakshirajan ◽  
M. Sreehari
Keyword(s):  
Markov Process ◽  
Iterated Logarithm ◽  
The Law

scholarly journals Numerical Methods for the Exit Time of a Piecewise-Deterministic Markov Process

2012 ◽  
Vol 44 (01) ◽  
pp. 196-225 ◽  
Author(s):  
Adrien Brandejsky ◽  
Benoîte De Saporta ◽  
François Dufour
Keyword(s):  
Markov Chain ◽  
Numerical Methods ◽  
Markov Process ◽  
Numerical Method ◽  
Rate Of Convergence ◽  
Discrete Time ◽  
Exit Time ◽  
Survival Function ◽  
Piecewise Deterministic Markov Process ◽  
Discrete Time Markov Chain

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

scholarly journals Numerical Methods for the Exit Time of a Piecewise-Deterministic Markov Process

2012 ◽  
Vol 44 (1) ◽  
pp. 196-225 ◽  
Author(s):  
Adrien Brandejsky ◽  
Benoîte De Saporta ◽  
François Dufour
Keyword(s):  
Markov Chain ◽  
Numerical Methods ◽  
Markov Process ◽  
Numerical Method ◽  
Rate Of Convergence ◽  
Discrete Time ◽  
Exit Time ◽  
Survival Function ◽  
Piecewise Deterministic Markov Process ◽  
Discrete Time Markov Chain

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

scholarly journals THE LAW OF LARGE NUMBERS AND THE LAW OF THE ITERATED LOGARITHM FOR INFINITE DIMENSIONAL INTERACTING DIFFUSION PROCESSES

2003 ◽  
Vol 06 (03) ◽  
pp. 489-503 ◽  
Author(s):  
BYRON SCHMULAND ◽  
WEI SUN
Keyword(s):  
Markov Process ◽  
Large Scale ◽  
Law Of Large Numbers ◽  
Diffusion Processes ◽  
Gibbs Measures ◽  
Exceptional Sets ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Scale Properties

The classical Dirichlet form given by the intrinsic gradient on Γℝd is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasi-sure analysis of Dirichlet forms is used to find exceptional sets of configurations for this Markov process. We consider large scale properties of the configuration and show that, for quite general measures, the process never hits those unusual configurations that violate the law of large numbers. Furthermore, for certain Gibbs measures, which model random particles in ℝd that interact via a potential function, we show, for d=1, 2, that the process never hits those unusual configurations that violate the law of the iterated logarithm.

Extreme values and the law of the iterated logarithm

1987 ◽  
Vol 74 (3) ◽  
pp. 319-340 ◽  
Author(s):  
J. Kuelbs ◽  
M. Ledoux
Keyword(s):  
Extreme Values ◽  
Iterated Logarithm ◽  
The Law

The law of the Iterated Logarithm for random interval homeomorphisms

2021 ◽  
Author(s):  
Klaudiusz Czudek ◽  
Tomasz Szarek ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Random Interval ◽  
Iterated Logarithm ◽  
The Law

On the Limit Set in the Law of the Iterated Logarithm for U-statistics of Order Two

2004 ◽  
pp. 111-126
Author(s):  
Stanislaw Kwapień ◽  
Rafał Latała ◽  
Krzysztof Oleszkiewicz ◽  
Joel Zinn
Keyword(s):  
Limit Set ◽  
U Statistics ◽  
Iterated Logarithm ◽  
The Law
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