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This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time…
We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive…
We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class…
We present an approach for testing for the existence of continuous generators of discrete stochastic transition matrices. Typically, the known approaches to ascertain the existence of continuous Markov processes are based in the assumption…
Stochastic convergence of discrete time Markov processes has been analysed based on a dual Lyapunov approach. Using some existing results on ergodic theory of Markov processes, it has been shown that existence of a properly subinvariant…
For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all…
We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with…
The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system ``intersect'' in finite time with probability one, is extended to cover situations when the coupling is unnecessarily…
We have studied Markov processes on denumerable state space and continuous time. We found that all these processes are connected via gauge transformations. We have used this result before as a method for resolution of equations, included…
Piecewise Deterministic Markov Processes (PDMPs) are studied in a general framework. First, different constructions are proven to be equivalent. Second, we introduce a coupling between two PDMPs following the same differential flow which…
We consider a general honest homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such…
Changing time of simple continuous-time Markov counting processes by independent unit-rate Poisson processes results in Markov counting processes for which we provide closed-form transition rates via composition of trajectories and with…
In this paper, we study a notion of local stationarity for discrete time Markov chains which is useful for applications in statistics. In the spirit of some locally stationary processes introduced in the literature, we consider triangular…
Following on from the notion of (first-order) causality, which generalises the notion of being tracepreserving from CP-maps to abstract processes, we give a characterization for the most general kind of map which sends causal processes to…
We provide a coupling proof of Doob's theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure $\mu$ converge to $\mu$ in the total variation distance. In addition we show…
We observe the continuous-time Markov Branching Process without high-order moments and allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. Main mathematical tool is…
Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample…
We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same…
In this paper we study the asymptotic behavior of linear processes having as innovations mean zero, square integrable functions of stationary reversible Markov chains. In doing so we shall preserve the generality of coefficients assuming…
Comparison results are given for time-inhomogeneous Markov processes with respect to function classes induced stochastic orderings. The main result states comparison of two processes, provided that the comparability of their infinitesimal…