Stochastic domination and Markovian couplings

2000 ◽  
Vol 32 (4) ◽  
pp. 1064-1076 ◽  
Author(s):  
F. Javier López ◽  
Servet Martínez ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Stochastic Domination ◽  
Jackson Networks ◽  
Partially Ordered ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

Lumpability and marginalisability for continuous-time Markov chains

1993 ◽  
Vol 30 (3) ◽  
pp. 518-528 ◽  
Author(s):  
Frank Ball ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Ion Channel ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Continuous Time Markov Chains ◽  
Probabilistic Proof ◽  
Necessary And Sufficient ◽  
Mutually Independent

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (01) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

On the Problem of Establishing the Existence of Stationary Distributions for Continuous-Time Markov Chains

1993 ◽  
Vol 7 (4) ◽  
pp. 529-543 ◽  
Author(s):  
P. K. Pollett ◽  
P. G. Taylor
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Stationary Distributions ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient ◽  
Networks Of Queues

We consider the problem of establishing the existence of stationary distributions for continuous-time Markov chains directly from the transition rates Q. Given an invariant probability distribution m for Q, we show that a necessary and sufficient condition for m to be a stationary distribution for the minimal process is that Q be regular. We provide sufficient conditions for the regularity of Q that are simple to verify in practice, thus allowing one to easily identify stationary distributions for a variety of models. To illustrate our results, we shall consider three classes of multidimensional Markov chains, namely, networks of queues with batch movements, semireversible queues, and partially balanced Markov processes.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (1) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Classification of involutions on finitary incidence algebras

2014 ◽  
Vol 24 (08) ◽  
pp. 1085-1098 ◽  
Author(s):  
Rosali Brusamarello ◽  
Érica Zancanella Fornaroli ◽  
Ednei Aparecido Santulo
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Decomposition Theorem ◽  
Multilinear Algebra ◽  
Partially Ordered ◽  
Incidence Algebras ◽  
Linear Multilinear Algebra ◽  
Necessary And Sufficient

Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI (X), to be equivalent in the case when every multiplicative automorphism of FI (X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra 60 (2012) 181–188] for finitary incidence algebras.

On geometric and algebraic transience for block-structured Markov chains

2020 ◽  
Vol 57 (4) ◽  
pp. 1313-1338
Author(s):  
Yuanyuan Liu ◽  
Wendi Li ◽  
Xiuqin Li
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stability Properties ◽  
System Parameters ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient ◽  
Block Structured

AbstractBlock-structured Markov chains model a large variety of queueing problems and have many important applications in various areas. Stability properties have been well investigated for these Markov chains. In this paper we will present transient properties for two specific types of block-structured Markov chains, including M/G/1 type and GI/M/1 type. Necessary and sufficient conditions in terms of system parameters are obtained for geometric transience and algebraic transience. Possible extensions of the results to continuous-time Markov chains are also included.

Lumpability and marginalisability for continuous-time Markov chains

1993 ◽  
Vol 30 (03) ◽  
pp. 518-528 ◽  
Author(s):  
Frank Ball ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Ion Channel ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Continuous Time Markov Chains ◽  
Probabilistic Proof ◽  
Necessary And Sufficient ◽  
Mutually Independent

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X 1(t), X 2(t), · ··, Xm (t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X 1(t)}, {X 2(t)}, · ··, {Xm (t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

The Periodic Radical of Group Rings and Incidence Algebras

1998 ◽  
Vol 41 (4) ◽  
pp. 481-487 ◽  
Author(s):  
M. M. Parmenter ◽  
E. Spiegel ◽  
P. N. Stewart
Keyword(s):  
Group Ring ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Locally Finite ◽  
Partially Ordered ◽  
Incidence Algebras ◽  
Finite Partially Ordered Set ◽  
Necessary And Sufficient

AbstractLet R be a ring with 1 and P(R) the periodic radical of R. We obtain necessary and sufficient conditions for P(RG) = 0 when RG is the group ring of an FC group G and R is commutative. We also obtain a complete description of when I(X, R) is the incidence algebra of a locally finite partially ordered set X and R is commutative.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE STOCHASTIC COMPARISON OF JACKSON NETWORKS

2003 ◽  
Vol 17 (1) ◽  
pp. 143-151 ◽  
Author(s):  
Antonis Economou
Keyword(s):  
Continuous Time ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Comparison ◽  
Jackson Networks ◽  
Continuous Time Markov Chains ◽  
Service Rates ◽  
Necessary And Sufficient

External and internal monotonicity properties for Jackson networks have been established in the literature with the use of coupling constructions. Recently, Lopez et al. derived necessary and sufficient conditions for the (strong) stochastic comparison of two-station Jackson networks with increasing service rates, by constructing a certain Markovian coupling. In this article, we state necessary and sufficient conditions for the stochastic comparison of L-station Jackson networks in the general case. The proof is based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

Conditions for strong ergodicity using intensity matrices

1988 ◽  
Vol 25 (1) ◽  
pp. 34-42 ◽  
Author(s):  
Jean Johnson ◽  
Dean Isaacson
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Discrete Time ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Limiting Behavior ◽  
Strong Ergodicity ◽  
Continuous Time Markov Chains

Sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given in several papers. Conditions have been given using the left eigenvectors ψn of Pn(ψ nPn = ψ n) and also using the limiting behavior of Pn. In this paper we consider the analogous results in the case of continuous-time Markov chains where one uses the intensity matrices Q(t) instead of P(s, t). A bound on the rate of convergence of certain strongly ergodic chains is also given.

Sign in / Sign up

Export Citation Format

Share Document