The square root law in the embedding detection problem for Markov chains with unknown matrix of transition probabilities

2019 ◽  
Vol 29 (1) ◽  
pp. 59-68
Author(s):  
Artem V. Volgin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Classical Model ◽  
Transition Probability Matrix ◽  
Square Root ◽  
Detection Problem ◽  
Asymptotic Growth

Abstract We consider the classical model of embeddings in a simple binary Markov chain with unknown transition probability matrix. We obtain conditions on the asymptotic growth of lengths of the original and embedded sequences sufficient for the consistency of the proposed statistical embedding detection test.

scholarly journals Infinitely divisible random transition probabilities with application to dependent Markov chains

1984 ◽  
Vol 25 (4) ◽  
pp. 463-472
Author(s):  
Peter L. Chesson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
White Noise ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Rates ◽  
Infinitely Divisible ◽  
Random Transition ◽  
Transition Probability Matrices ◽  
Probability Matrices

AbstractRandom transition probability matrices with stationary independent factors define “white noise” environment processes for Markov chains. Two examples are considered in detail. Such environment processes can be used to construct several Markov chains which are dependent, have the same transition probabilities and are jointly a Markov chain. Transition rates for such processes are evaluated. These results have application to the study of animal movements.

Some explicit results for correlated random walks

1989 ◽  
Vol 26 (4) ◽  
pp. 757-766 ◽  
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Equilibrium Distribution ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Correlated Random Walks ◽  
Finite State ◽  
Direction Of Movement ◽  
Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

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.

Some explicit results for correlated random walks

1989 ◽  
Vol 26 (04) ◽  
pp. 757-766 ◽  
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Equilibrium Distribution ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Correlated Random Walks ◽  
Finite State ◽  
Direction Of Movement ◽  
Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

scholarly journals A geometric invariant in weak lumpability of finite Markov chains

1997 ◽  
Vol 34 (4) ◽  
pp. 847-858 ◽  
Author(s):  
James Ledoux
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Spectral Properties ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Homogeneous Markov Chain ◽  
Geometric Invariant ◽  
Initial State ◽  
Direct Sum ◽  
Finite Markov Chains

We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.

Weak lumpability in finite Markov chains

1982 ◽  
Vol 19 (03) ◽  
pp. 685-691 ◽  
Author(s):  
Atef M. Abdel-moneim ◽  
Frederick W. Leysieffer
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Transition Probability ◽  
Linear Equations ◽  
Transition Probability Matrix ◽  
Finite Markov Chain ◽  
Systems Of Linear Equations ◽  
Finite Markov Chains ◽  
The Given

Criteria are given to determine whether a given finite Markov chain can be lumped weakly with respect to a given partition of its state space. These conditions are given in terms of solution classes of systems of linear equations associated with the transition probability matrix of the Markov chain and the given partition.

Weak lumpability in finite Markov chains

1982 ◽  
Vol 19 (3) ◽  
pp. 685-691 ◽  
Author(s):  
Atef M. Abdel-moneim ◽  
Frederick W. Leysieffer
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Transition Probability ◽  
Linear Equations ◽  
Transition Probability Matrix ◽  
Finite Markov Chain ◽  
Systems Of Linear Equations ◽  
Finite Markov Chains ◽  
The Given

Criteria are given to determine whether a given finite Markov chain can be lumped weakly with respect to a given partition of its state space. These conditions are given in terms of solution classes of systems of linear equations associated with the transition probability matrix of the Markov chain and the given partition.

Investigation of Numerical Performance of Partitioning and Parallel Processing of Markov Chain (PPMC) for Complex Design Problems

2014 ◽  
Author(s):  
Po Ting Lin ◽  
Yu-Cheng Chou ◽  
Mark Christian E. Manuel ◽  
Kuan Sung Hsu
Keyword(s):  
Markov Chain ◽  
Parallel Processing ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Divide And Conquer ◽  
Threshold Parameter ◽  
Network Systems ◽  
Numerical Efficiency ◽  
Numerical Performance

Divide-and-conquer strategies have been utilized to perform evaluation calculations of complex network systems, such as reliability analysis of a Markov chain. This paper focuses on partitioning of Markov chain for a multi-modular redundant system and the fast calculation using parallel processing. The complexity of Markov chain is first reduced by eliminating the connections with low transition probabilities associated with a threshold parameter. The transition probability matrix is then reordered and partitioned such that a worse-case reliability is evaluated through the calculations in only the diagonal sub-matrices of the transition probability matrix. Since the calculations of the sub-matrices are independent to each other, the numerical efficiency can be greatly improved using parallel computing. The numerical results showed the selection of threshold parameter is a key factor to numerical efficiency. In this paper, the sensitivity of the numerical performance of Partitioning and Parallel-processing of Markov Chain (PPMC) to the threshold parameter has been investigated and discussed.

scholarly journals A geometric invariant in weak lumpability of finite Markov chains

1997 ◽  
Vol 34 (04) ◽  
pp. 847-858 ◽  
Author(s):  
James Ledoux
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Spectral Properties ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Homogeneous Markov Chain ◽  
Geometric Invariant ◽  
Initial State ◽  
Direct Sum ◽  
Finite Markov Chains

We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.

Further enhancement on H∞ sliding mode control design for singular Markovian jump systems with partly known transition probabilities

2021 ◽  
pp. 107754632198920
Author(s):  
Zeinab Fallah ◽  
Mahdi Baradarannia ◽  
Hamed Kharrati ◽  
Farzad Hashemzadeh
Keyword(s):  
Transition Probabilities ◽  
Sliding Mode ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Linear Matrix ◽  
Sliding Mode Controller ◽  
Sliding Surface ◽  
Markovian Jump ◽  
Markovian Jump System ◽  
Singular Markovian Jump Systems

This study considers the designing of the H ∞ sliding mode controller for a singular Markovian jump system described by discrete-time state-space realization. The system under investigation is subject to both matched and mismatched external disturbances, and the transition probability matrix of the underlying Markov chain is considered to be partly available. A new sufficient condition is developed in terms of linear matrix inequalities to determine the mode-dependent parameter of the proposed quasi-sliding surface such that the stochastic admissibility with a prescribed H ∞ performance of the sliding mode dynamics is guaranteed. Furthermore, the sliding mode controller is designed to assure that the state trajectories of the system will be driven onto the quasi-sliding surface and remain in there afterward. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design algorithms.

Sign in / Sign up

Export Citation Format

Share Document