Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (4) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn, where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and πn for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of πn to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r0 > 1 with , then the exact convergence rate of πn to π is characterized by r0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between πn and π based on the moments of A.

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

On the probability generating function of the sum of Markov Bernoulli random variables

1982 ◽  
Vol 19 (A) ◽  
pp. 321-326 ◽  
Author(s):  
J. Gani
Keyword(s):  
Markov Chain ◽  
Generating Function ◽  
Transition Probability ◽  
Probability Generating Function ◽  
Random Variables ◽  
Direct Proof ◽  
Transition Probability Matrix ◽  
Bernoulli Random Variables ◽  
Limit Probability

A direct proof of the expression for the limit probability generating function (p.g.f.) of the sum of Markov Bernoulli random variables is outlined. This depends on the larger eigenvalue of the transition probability matrix of their Markov chain.

scholarly journals Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices

2016 ◽  
Vol 53 (3) ◽  
pp. 946-952
Author(s):  
Loï Hervé ◽  
James Ledoux
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Spectral Radius ◽  
Spectral Gap ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Geometric Ergodicity ◽  
Transition Matrices ◽  
Essential Spectral Radius ◽  
Band Transition

AbstractWe analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

scholarly journals On the Convergence Rate of Bonus-Malus Systems

1992 ◽  
Vol 22 (2) ◽  
pp. 217-223 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Convergence Rate ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Limit Probability ◽  
Step Transition

AbstractUnder certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

On the move-to-front scheme with Markov dependent requests

1997 ◽  
Vol 34 (3) ◽  
pp. 790-794 ◽  
Author(s):  
R. M. Phatarfod ◽  
A. J. Pryde ◽  
David Dyte
Keyword(s):  
Markov Chain ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Transition Probability Matrix ◽  
Principal Submatrices

In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.

On the move-to-front scheme with Markov dependent requests

1997 ◽  
Vol 34 (03) ◽  
pp. 790-794 ◽  
Author(s):  
R. M. Phatarfod ◽  
A. J. Pryde ◽  
David Dyte
Keyword(s):  
Markov Chain ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Transition Probability Matrix ◽  
Principal Submatrices

In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P . It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.

Nonhomogeneous Stochastic Automata

1981 ◽  
Vol 4 (4) ◽  
pp. 891-917
Author(s):  
Sławomir Janicki
Keyword(s):  
Transition Probability ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Initial Distribution ◽  
Transition Probability Matrix ◽  
Extension Problem ◽  
Stochastic Automaton ◽  
Nonhomogeneous Markov Chain ◽  
Matrix Sequence ◽  
Necessary And Sufficient

In this note we consider a nonhomogeneous Markov chain type stochastic automaton which is a generalization of Bartoszyński’s stochastic automaton. The latter is a generalization of the Pawlak’s known machine in a stochastic direction. By nonhomogeneous stochastic automaton we mean a system ⟨ T, α, {A(n), n ⩾ 1}⟩, where T is a finite nonempty set, α, is an initial distribution on T, and {A(n), n ⩾ 1} is a matrix sequence whose every element is a stochastic matrix called a transition probability matrix. If A(n) = A for all n ⩾ 1, then we obtain Bartoszyński’s automaton. The sequence (ti0, ti1, …), tij ∈ T, j = 0, 1, 2, … is called a word of automata if α(ti0) > 0 and A(k)(tik-1, tik) > 0 for every k ⩾ 1. The goal of this note is to give necessary and sufficient conditions for the existence of an extension and a shrinkage of the automata under consideration. These problems for T, A were considered for the first time by Bartoszyński. The shrinkage problem deals with the existence of a stochastic automaton which generates only all sequences of states of T which are simultaneously generated by two given automata while the extension problem treats of the existence of a stochastic automaton which generates all sequences of states of which are generated by at least one of two given automata. Moreover, we introduce some new notions: attainable state, concordance of automata in a wide and a narrow sense, which help us to solve the problems mentioned above.

On the probability generating function of the sum of Markov Bernoulli random variables

1982 ◽  
Vol 19 (A) ◽  
pp. 321-326 ◽  
Author(s):  
J. Gani
Keyword(s):  
Markov Chain ◽  
Generating Function ◽  
Transition Probability ◽  
Probability Generating Function ◽  
Random Variables ◽  
Direct Proof ◽  
Transition Probability Matrix ◽  
Bernoulli Random Variables ◽  
Limit Probability

A direct proof of the expression for the limit probability generating function (p.g.f.) of the sum of Markov Bernoulli random variables is outlined. This depends on the larger eigenvalue of the transition probability matrix of their Markov chain.

scholarly journals A Regression-Based Procedure for Markov Transition Probability Estimation in Land Change Modeling

Land ◽  
2020 ◽  
Vol 9 (11) ◽  
pp. 407
Author(s):  
J. Ronald Eastman ◽  
Jiena He
Keyword(s):  
Transition Probabilities ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Maximum Error ◽  
Transition Probability Matrix ◽  
Approximation Technique ◽  
Land Change ◽  
Change Models ◽  
Markov Transition ◽  
Land Change Modeling

Land change models commonly model the expected quantity of change as a Markov chain. Markov transition probabilities can be estimated by tabulating the relative frequency of change for all transitions between two dates. To estimate the appropriate transition probability matrix for any future date requires the determination of an annualized matrix through eigendecomposition followed by matrix powering. However, the technique yields multiple solutions, commonly with imaginary parts and negative transitions, and possibly with no non-negative real stochastic matrix solution. In addition, the computational burden of the procedure makes it infeasible for practical use with large problems. This paper describes a Regression-Based Markov (RBM) approximation technique based on quadratic regression of individual transitions that is shown to always yield stochastic matrices, with very low error characteristics. Using land cover data for the 48 conterminous US states, median errors in probability for the five states with the highest rates of transition were found to be less than 0.00001 and the maximum error of 0.006 was of the same order of magnitude experienced by the commonly used compromise of forcing small negative transitions estimated by eigendecomposition to 0. Additionally, the technique can solve land change modeling problems of any size with extremely high computational efficiency.

Stability Analysis of the Nanjung Water Diversion Twin Tunnels based on Convergence Measurement

2019 ◽  
Vol 1 (1) ◽  
pp. 49-60
Author(s):  
Simon Heru Prassetyo ◽  
Ganda Marihot Simangunsong ◽  
Ridho Kresna Wattimena ◽  
Made Astawa Rai ◽  
Irwandy Arif ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Convergence Rate ◽  
Critical Strain ◽  
Convergence Rates ◽  
Strain Analysis ◽  
Water Diversion ◽  
Tunnel Face ◽  
Tunnel Stability ◽  
The Stability ◽  
Twin Tunnels

This paper focuses on the stability analysis of the Nanjung Water Diversion Twin Tunnels using convergence measurement. The Nanjung Tunnel is horseshoe-shaped in cross-section, 10.2 m x 9.2 m in dimension, and 230 m in length. The location of the tunnel is in Curug Jompong, Margaasih Subdistrict, Bandung. Convergence monitoring was done for 144 days between February 18 and July 11, 2019. The results of the convergence measurement were recorded and plotted into the curves of convergence vs. day and convergence vs. distance from tunnel face. From these plots, the continuity of the convergence and the convergence rate in the tunnel roof and wall were then analyzed. The convergence rates from each tunnel were also compared to empirical values to determine the level of tunnel stability. In general, the trend of convergence rate shows that the Nanjung Tunnel is stable without any indication of instability. Although there was a spike in the convergence rate at several STA in the measured span, that spike was not replicated by the convergence rate in the other measured spans and it was not continuous. The stability of the Nanjung Tunnel is also confirmed from the critical strain analysis, in which most of the STA measured have strain magnitudes located below the critical strain line and are less than 1%.

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