A note on roots of Markov shifts

1976 ◽  
Vol 13 (3) ◽  
pp. 591-596
Author(s):  
S. M. Rudolfer
Keyword(s):  
Markov Chain ◽  
Shift Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Markov Shifts ◽  
Necessary And Sufficient

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.

A note on roots of Markov shifts

1976 ◽  
Vol 13 (03) ◽  
pp. 591-596
Author(s):  
S. M. Rudolfer
Keyword(s):  
Markov Chain ◽  
Shift Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Markov Shifts ◽  
Necessary And Sufficient

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (02) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima M n of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants B n such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions π i=1 m H i (x). The effect of each factor H i (x) on the behavior of maxima from π i=1 m H i is analyzed. Under a mild regularity condition, B n can be chosen to be the maximum of the m quantiles of order (1 - n -1) of the H's.

Almost sure stability of maxima

1973 ◽  
Vol 10 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

Almost sure stability of maxima

1973 ◽  
Vol 10 (02) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n≧ 1} unbounded above setMn= max {X1,X2, …,Xn}. When do normalizing constantsbnexist such thatMn/bn→1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for alland in this casebn∼F–1(1 – 1/n) Necessary and sufficient conditions for lim supn→∞,Mn/bn= l >1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn= 1 a.s. except whenl= 1. When theXnare r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

scholarly journals Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities

1978 ◽  
Vol 15 (4) ◽  
pp. 848-851 ◽  
Author(s):  
Jean-François Mertens ◽  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Of A Solution ◽  
Necessary And Sufficient ◽  
Irreducible Markov Chain

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk < yo, · · ·, yN–1 for some k ≧ N, is necessary and sufficient for transience.

Stability in a non-homogeneous Markov chain model in manpower systems

1981 ◽  
Vol 18 (04) ◽  
pp. 924-930 ◽  
Author(s):  
P.-C. G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Homogeneous Markov Chain ◽  
Initial Structure ◽  
Chain Model ◽  
Limiting Behaviour ◽  
Necessary And Sufficient

Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability imposed on the initial structure and keep the same sequence of expansion. Finally we examine the limiting behaviour of the expected grade sizes if we drop the assumption of a continuously expanding system.

Limiting second moments for transient states of Markov chains

1973 ◽  
Vol 10 (04) ◽  
pp. 891-894
Author(s):  
H. P. Wynn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Time Points ◽  
Transient States ◽  
Second Moments ◽  
Necessary And Sufficient

The set of transient states of a Markov chain is considered as a system. If numbers of arrivals to the system at discrete time points have constant mean and covariance matrix then there is a limiting distribution of numbers in the states. Necessary and sufficient conditions are given for this distribution to yield zero correlations between states.

Transience and recurrence of an interesting Markov chain

1974 ◽  
Vol 11 (04) ◽  
pp. 818-824 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chain ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Recurrence ◽  
Null Recurrence ◽  
A Chain ◽  
Necessary And Sufficient ◽  
Countable Case

This note studies the natural extension to the countable case of a chain considered by Hendricks (1972) and gives necessary and sufficient conditions for transience, null recurrence and positive recurrence.

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