scholarly journals Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

2003 ◽  
Vol 7 ◽  
pp. 115-146 ◽  
Author(s):  
Jean-Pierre Conze ◽  
Albert Raugi
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
Transfer Operator

scholarly journals A Fourth Moment Inequality for Functionals of Stationary Processes

2008 ◽  
Vol 45 (04) ◽  
pp. 1086-1096
Author(s):  
Olivier Durieu
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
Invariance Principle ◽  
Spectral Properties ◽  
Empirical Process ◽  
Transfer Operator ◽  
Partial Sums ◽  
Stationary Processes ◽  
Fourth Moment ◽  
Moment Inequality

In this paper, a fourth moment bound for partial sums of functionals of strongly ergodic Markov chains is established. This type of inequality plays an important role in the study of the empirical process invariance principle. This inequality is specially adapted to the technique of Dehling, Durieu, and Volný (2008). The same moment bound can be proved for dynamical systems whose transfer operator has some spectral properties. Examples of applications are given.

scholarly journals A Fourth Moment Inequality for Functionals of Stationary Processes

2008 ◽  
Vol 45 (4) ◽  
pp. 1086-1096 ◽  
Author(s):  
Olivier Durieu
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
Invariance Principle ◽  
Spectral Properties ◽  
Empirical Process ◽  
Transfer Operator ◽  
Partial Sums ◽  
Stationary Processes ◽  
Fourth Moment ◽  
Moment Inequality

In this paper, a fourth moment bound for partial sums of functionals of strongly ergodic Markov chains is established. This type of inequality plays an important role in the study of the empirical process invariance principle. This inequality is specially adapted to the technique of Dehling, Durieu, and Volný (2008). The same moment bound can be proved for dynamical systems whose transfer operator has some spectral properties. Examples of applications are given.

scholarly journals Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

2022 ◽  
pp. 1-47
Author(s):  
Amarjit Budhiraja ◽  
Nicolas Fraiman ◽  
Adam Waterbury
Keyword(s):  
Markov Chain ◽  
Dynamical Systems ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Stochastic Dynamical Systems ◽  
Stationary Distributions ◽  
Small Noise ◽  
Positive Orthant ◽  
Limit Points

Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

scholarly journals Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

2018 ◽  
Vol 170 (6) ◽  
pp. 1051-1087 ◽  
Author(s):  
Fadi Antown ◽  
Davor Dragičević ◽  
Gary Froyland
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
Optimal Linear ◽  
Stochastically Perturbed

scholarly journals On the Computation of Invariant Measures in Random Dynamical Systems

2003 ◽  
Vol 03 (02) ◽  
pp. 247-265 ◽  
Author(s):  
Peter Imkeller ◽  
Peter Kloeden
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
State Space ◽  
Difference Equations ◽  
Invariant Measures ◽  
Random Dynamical Systems ◽  
Continuum State ◽  
Transition Mechanism ◽  
Definition Of ◽  
Stationary Random Processes

Invariant measures of dynamical systems generated e.g. by difference equations can be computed by discretizing the originally continuum state space, and replacing the action of the generator by the transition mechanism of a Markov chain. In fact they are approximated by stationary vectors of these Markov chains. Here we extend this well-known approximation result and the underlying algorithm to the setting of random dynamical systems, i.e. dynamical systems on the skew product of a probability space carrying the underlying stationary stochasticity and the state space, a particular non-autonomous framework. The systems are generated by difference equations driven by stationary random processes modelled on a metric dynamical system. The approximation algorithm involves spatial discretizations and the definition of appropriate random Markov chains with stationary vectors converging to the random invariant measure of the system.

scholarly journals Large and moderate deviations for bounded functions of slowly mixing Markov chains

2017 ◽  
Vol 18 (02) ◽  
pp. 1850017
Author(s):  
Jérôme Dedecker ◽  
Sébastien Gouëzel ◽  
Florence Merlevède
Keyword(s):  
Probability Theory ◽  
Dynamical Systems ◽  
Markov Chains ◽  
Weak Sense ◽  
Moderate Deviations ◽  
Mixing Speed ◽  
Bounded Functions

We consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities. These inequalities can be applied in various natural situations coming from probability theory or dynamical systems. Finally, we discuss examples from these various settings showing that our inequalities are sharp.

Sign in / Sign up

Export Citation Format

Share Document