Invariance principles for the empirical measure of a mixing sequence and for the local time of markov processes

1986 ◽  
pp. 4-21
Author(s):  
P. Doukhan ◽  
J. R. Leon
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Empirical Measure ◽  
Invariance Principles ◽  
Mixing Sequence

The local time of the Markov processes of Ornstein–Uhlenbeck type

2012 ◽  
Vol 82 (7) ◽  
pp. 1229-1234
Author(s):  
Changqing Tong ◽  
Zhengyan Lin ◽  
Jing Zheng
Keyword(s):  
Local Time ◽  
Markov Processes

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.

Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

2013 ◽  
Vol 50 (04) ◽  
pp. 931-942
Author(s):  
Takayuki Fujii
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Nonparametric Estimation ◽  
Sample Path ◽  
Point Of View ◽  
Statistical Point ◽  
Jump Size ◽  
Stationary Density ◽  
Piecewise Deterministic Markov Processes ◽  
Estimation Problems

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

scholarly journals Markov processes with spatial delay: Path space characterization, occupation time and properties

2017 ◽  
Vol 17 (06) ◽  
pp. 1750042 ◽  
Author(s):  
Michael Salins ◽  
Konstantinos Spiliopoulos
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Single Point ◽  
Occupation Time ◽  
One Dimensional ◽  
Stochastic Dynamical System ◽  
Second Derivatives ◽  
Continuous Markov Process ◽  
Positive Time ◽  
Definition Of

In this paper, we study one-dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one-dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator’s domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the generator has restrictions involving second derivatives. In this paper we provide a pathwise characterization for processes with delay in terms of an SDE and an occupation time formula involving the symmetric local time. This characterization provides an explicit Doob–Meyer decomposition, demonstrating that such processes are semi-martingales and that all of stochastic calculus including Itô formula and Girsanov formula applies. We also establish an occupation time formula linking the time that the process spends at a delay point with its symmetric local time there. A physical example of a stochastic dynamical system with delay is lastly presented and analyzed.

scholarly journals Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

2013 ◽  
Vol 50 (4) ◽  
pp. 931-942
Author(s):  
Takayuki Fujii
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Nonparametric Estimation ◽  
Sample Path ◽  
Point Of View ◽  
Statistical Point ◽  
Jump Size ◽  
Stationary Density ◽  
Piecewise Deterministic Markov Processes ◽  
Estimation Problems

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

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