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.

Approximation of the invariant probability measure of an infinite stochastic matrix

1980 ◽  
Vol 12 (03) ◽  
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 {P m }m∊N of stochastic matrices converging to P (pointwise), such that every Pm has at least one invariant probability measure π m . The aim of this paper is to find conditions, which assure that at least one of sequences {π m }m∊N converges to π (pointwise). This includes the case where the P m 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 {π m }m∊N converges to π. The conditions given for the general case are closely related to Foster's condition.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

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).

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

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 &gt; 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| &gt; 1, we derive an upper bound for the distance between π n and π based on the moments of A.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

2018 ◽  
Vol 50 (2) ◽  
pp. 645-669 ◽  
Author(s):  
Yuanyuan Liu ◽  
Wendi Li
Keyword(s):  
Markov Chains ◽  
Perturbation Method ◽  
Error Bounds ◽  
Continuous Time ◽  
Stochastic Matrix ◽  
Probability Vector ◽  
The Poisson Equation ◽  
Residual Matrix ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

AbstractLetPbe the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vectorπT, and let(n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n+ 1) x (n+ 1) northwest corner truncation ofParbitrarily, with invariant probability vector(n)πT. We derive computableV-norm bounds on the error betweenπTand(n)πTin terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 asntends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

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.

Continuity of Lyapunov exponents for random two-dimensional matrices

2016 ◽  
Vol 37 (5) ◽  
pp. 1413-1442 ◽  
Author(s):  
CARLOS BOCKER-NETO ◽  
MARCELO VIANA
Keyword(s):  
Probability Measure ◽  
Lyapunov Exponents ◽  
Invariant Probability Measure ◽  
Two Dimensional ◽  
Bernoulli Shifts

The Lyapunov exponents of locally constant$\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.

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