scholarly journals A note on repeated sequences in Markov chains

1987 ◽  
Vol 19 (3) ◽  
pp. 739-742 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Transition Matrix ◽  
Repeated Sequences ◽  
Markov Renewal Process ◽  
Conditional Mean ◽  
Sojourn Times

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

scholarly journals A note on repeated sequences in Markov chains

1987 ◽  
Vol 19 (03) ◽  
pp. 739-742 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Transition Matrix ◽  
Repeated Sequences ◽  
Markov Renewal Process ◽  
Conditional Mean ◽  
Sojourn Times

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

scholarly journals Relative stability and weak convergence in non-decreasing stochastically monotone Markov chains

1991 ◽  
Vol 4 (4) ◽  
pp. 293-303
Author(s):  
P. Todorovic
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Renewal Process ◽  
Relative Stability ◽  
Transition Probability ◽  
Markov Renewal Process

Let {ξn} be a non-decreasing stochastically monotone Markov chain whose transition probability Q(.,.) has Q(x,{x})=β(x)>0 for some function β(.) that is non-decreasing with β(x)↑1 as x→+∞, and each Q(x,.) is non-atomic otherwise. A typical realization of {ξn} is a Markov renewal process {(Xn,Tn)}, where ξj=Xn, for Tn consecutive values of j, Tn geometric on {1,2,…} with parameter β(Xn). Conditions are given for Xn, to be relatively stable and for Tn to be weakly convergent.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

scholarly journals Markov chains with exponential return times are finitary

2020 ◽  
pp. 1-9
Author(s):  
OMER ANGEL ◽  
YINON SPINKA
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Proper Subgroup ◽  
Ergodic Markov Chain ◽  
Countable State Space ◽  
Return Times ◽  
Exponential Tails ◽  
Stationary Version ◽  
Countable State

Abstract Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.

scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

scholarly journals On the Dual Relationship Between Markov Chains of GI/M/1 and M/G/1 Type

2010 ◽  
Vol 42 (1) ◽  
pp. 210-225 ◽  
Author(s):  
P. G. Taylor ◽  
B. Van Houdt
Keyword(s):  
Markov Chains ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Dual Processes ◽  
Design Of Algorithms ◽  
Type Process ◽  
The Matrix ◽  
Positive Recurrent ◽  
Matrix Transforms ◽  
Phd Thesis

In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

scholarly journals On the Embedding Problem for Discrete-Time Markov Chains

2013 ◽  
Vol 50 (04) ◽  
pp. 918-930 ◽  
Author(s):  
Marie-Anne Guerry
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Transition Matrix ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Embedding Problem ◽  
Time Intervals ◽  
Square Roots ◽  
Probability Matrices

When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable. In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

scholarly journals Estimating the Entropy Rate of Finite Markov Chains With Application to Behavior Studies

2019 ◽  
Vol 44 (3) ◽  
pp. 282-308 ◽  
Author(s):  
Brian G. Vegetabile ◽  
Stephanie A. Stout-Oswald ◽  
Elysia Poggi Davis ◽  
Tallie Z. Baram ◽  
Hal S. Stern
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Markov Processes ◽  
Simulation Study ◽  
Transition Matrix ◽  
Sliding Window ◽  
Entropy Rate ◽  
Homogeneous Markov Chain ◽  
The Third ◽  
Finite Markov Chains

Predictability of behavior is an important characteristic in many fields including biology, medicine, marketing, and education. When a sequence of actions performed by an individual can be modeled as a stationary time-homogeneous Markov chain the predictability of the individual’s behavior can be quantified by the entropy rate of the process. This article compares three estimators of the entropy rate of finite Markov processes. The first two methods directly estimate the entropy rate through estimates of the transition matrix and stationary distribution of the process. The third method is related to the sliding-window Lempel–Ziv compression algorithm. The methods are compared via a simulation study and in the context of a study of interactions between mothers and their children.

Markov renewal processes, counters and repeated sequences in Markov chains

1987 ◽  
Vol 19 (03) ◽  
pp. 521-545 ◽  
Author(s):  
J. D. Biggins ◽  
C. Cannings
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Restriction Enzymes ◽  
Repeated Sequences ◽  
Renewal Processes ◽  
Basic Process ◽  
Markov Renewal Processes ◽  
Specific Sequences

The theory of Markov renewal processes is applied to study the occurrence of specific sequences of states in a Markov chain. Çinlar&s (1969) results are used to study both the basic process, and that obtained when the overlap of sequences is not permitted, as in the theory of counters. These results are applied to the fragments formed when DNA is digested using one, or more, restriction enzymes.

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