Positive columns for stochastic matrices

1974 ◽  
Vol 11 (4) ◽  
pp. 829-835 ◽  
Author(s):  
Dean Isaacson ◽  
Richard Madsen
Keyword(s):  
Positive Column ◽  
Stochastic Matrix ◽  
The State ◽  
Stochastic Matrices

If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.

Positive columns for stochastic matrices

1974 ◽  
Vol 11 (04) ◽  
pp. 829-835 ◽  
Author(s):  
Dean Isaacson ◽  
Richard Madsen
Keyword(s):  
Positive Column ◽  
Stochastic Matrix ◽  
The State ◽  
Stochastic Matrices

If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

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.

Homogeneous Markov chains by bounded transition matrix

1994 ◽  
Vol 31 (2) ◽  
pp. 362-372
Author(s):  
D. J. Hartfiel
Keyword(s):  
Markov Chains ◽  
Efficient Algorithm ◽  
Transition Matrix ◽  
Stochastic Matrix ◽  
Convex Polytope ◽  
Stochastic Matrices ◽  
Work Done

Let A be a stochastic matrix and ε a positive number. We consider all stochastic matrices within ε of A and their corresponding stochastic eigenvectors. A convex polytope containing these vectors is described. An efficient algorithm for computing bounds on the components of these vectors is also given. The work is compared to previous such work done by the author and by Courtois and Semai.

Vector Monte Carlo stochastic matrix-based algorithms for large linear systems

2016 ◽  
Vol 22 (3) ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Linear Systems ◽  
Search Strategy ◽  
Stochastic Matrix ◽  
Stochastic Methods ◽  
Vector Representation ◽  
Stochastic Matrices ◽  
Short Article ◽  
Gradient Based ◽  
Matrix Vector ◽  
Large Linear Systems

AbstractIn this short article we suggest randomized scalable stochastic matrix-based algorithms for large linear systems. The idea behind these stochastic methods is a randomized vector representation of matrix iterations. In addition, to minimize the variance, it is suggested to use stochastic and double stochastic matrices for efficient randomized calculation of matrix iterations and a random gradient based search strategy. The iterations are performed by sampling random rows and columns only, thus avoiding not only matrix matrix but also matrix vector multiplications. Further improvements of the methods can be obtained through projections by a random gaussian matrix.

A decomposition theorem for infinite stochastic matrices

1995 ◽  
Vol 32 (04) ◽  
pp. 893-901 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Stochastic Matrix ◽  
Decomposition Theorem ◽  
Stochastic Matrices ◽  
First Passage ◽  
Mean First Passage Times ◽  
Expected Values ◽  
Lower Triangular ◽  
Infinite State ◽  
Steady State Probabilities ◽  
Passage Times

We prove that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I – P=(A – I)(B – S), where A is strictly upper-triangular, B is strictly lower-triangular, and S is diagonal. Moreover, the elements of A are expected values of random variables that we will specify, and the elements of B and S are probabilities of events that we will specify. The decomposition can be used to obtain steady-state probabilities, mean first-passage-times and the fundamental matrix.

Approximation of the invariant probability measure of an infinite stochastic matrix

1980 ◽  
Vol 12 (3) ◽  
pp. 710-726 ◽  
Author(s):  
D. Wolf
Keyword(s):  
Probability Measure ◽  
Special Interest ◽  
Stochastic Matrix ◽  
Invariant Probability Measure ◽  
Stochastic Matrices ◽  
Positive Recurrent

Let P denote an irreducible positive recurrent infinite stochastic matrix with the unique invariant probability measure π. We consider sequences {Pm}m∊N of stochastic matrices converging to P (pointwise), such that every Pm has at least one invariant probability measure πm. The aim of this paper is to find conditions, which assure that at least one of sequences {πm}m∊N converges to π (pointwise). This includes the case where the Pm are finite matrices, which is of special interest. It is shown that there is a sequence of finite stochastic matrices, which can easily be constructed, such that {πm}m∊N converges to π. The conditions given for the general case are closely related to Foster's condition.

A Relationship between Arbitrary Positive Matrices and Stochastic Matrices

1966 ◽  
Vol 18 ◽  
pp. 303-306 ◽  
Author(s):  
Richard Sinkhorn
Keyword(s):  
Stochastic Matrix ◽  
Diagonal Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Positive Matrices ◽  
Square Matrix

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.

A decomposition theorem for infinite stochastic matrices

1995 ◽  
Vol 32 (4) ◽  
pp. 893-901 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Stochastic Matrix ◽  
Decomposition Theorem ◽  
Stochastic Matrices ◽  
First Passage ◽  
Mean First Passage Times ◽  
Expected Values ◽  
Lower Triangular ◽  
Infinite State ◽  
Steady State Probabilities ◽  
Passage Times

We prove that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I – P=(A – I)(B – S), where A is strictly upper-triangular, B is strictly lower-triangular, and S is diagonal. Moreover, the elements of A are expected values of random variables that we will specify, and the elements of B and S are probabilities of events that we will specify. The decomposition can be used to obtain steady-state probabilities, mean first-passage-times and the fundamental matrix.

scholarly journals The $n$-Card Problem, Stochastic Matrices, and the Extreme Principle

10.37236/2444 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Justin H.C. Chan ◽  
Jonathan Jedwab
Keyword(s):  
Assignment Problem ◽  
Closed Interval ◽  
Travelling Salesman Problem ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Travelling Salesman ◽  
Linear Assignment Problem ◽  
Mathematical Problems ◽  
Repeated Use ◽  
Linear Assignment

The $n$-card problem is to determine the minimal intervals $[u,v]$ such that for every $n \times n$ stochastic matrix $A$ there is an $n \times n$ permutation matrix $P$ (depending on $A$) such that tr$(PA) \in [u,v]$. This problem is closely related to classical mathematical problems from industry and management, including the linear assignment problem and the travelling salesman problem. The minimal intervals for the $n$-card problem are known only for $n \le 4$.We introduce a new method of analysis for the $n$-card problem that makes repeated use of the Extreme Principle. We use this method to answer a question posed by Sands (2011), by showing that $[1,2]$ is a solution to the $n$-card problem for all $n \ge 2$. We also show that each closed interval of length $\frac{n}{n-1}$ contained in $[0,2)$ is a solution to the $n$-card problem for all $n \ge 2$.

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