The embedding problem for Markov chains with three states

1980 ◽  
Vol 87 (2) ◽  
pp. 285-294 ◽  
Author(s):  
Halina Frydman
Keyword(s):  
Markov Chains ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Parameter Family ◽  
Necessary And Sufficient Conditions ◽  
Embedding Problem ◽  
Stochastic Matrices ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Two Parameter

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matricessatisfyingand such thatThough extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.

scholarly journals The Condition of Regularity in Simple Markov Chains

1956 ◽  
Vol 9 (3) ◽  
pp. 387
Author(s):  
J Gani
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Latent Root ◽  
Necessary And Sufficient Conditions ◽  
A Chain ◽  
Necessary And Sufficient

The paper generalizes a proof, and outlines an alternative to it, for the well-known theorem on the conditions of regularity in a simple Markov chain; this is that the necessary and sufficient conditions for a chain to be regular are that the latent root 1 of the stochastic matrix for the chain must be simple, and the remaining roots have moduli less than 1.

Poisson theorem for non-homogeneous Markov chains

1989 ◽  
Vol 26 (03) ◽  
pp. 637-642 ◽  
Author(s):  
Janusz Pawłowski
Keyword(s):  
Markov Chains ◽  
Poisson Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compound Poisson Distribution ◽  
Convergence In Distribution ◽  
Compound Poisson ◽  
Necessary And Sufficient

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

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.

On the Hausdorff and Trigonometric Moment Problems

1961 ◽  
Vol 13 ◽  
pp. 454-461
Author(s):  
P. G. Rooney
Keyword(s):  
Moment Problem ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Moment Problems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that(1)For various collections K this problem has been solved—see (3, Chapter III)By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that(2)For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

Coefficient Regions for Univalent Trinomials

1980 ◽  
Vol 32 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Waniurski
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient

The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].THEOREM A (Dieudonné criterion). The polynomial1is univalent in |z| < 1 if and only if for every Θ in [0, π/2] the associated polynomial2does not vanish in |z| < 1. For Θ = 0, (2) is to be interpreted as the derivative of (1).The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the functionis regular in the unit disk.

Traces of Matrices of Zeros and Ones

1960 ◽  
Vol 12 ◽  
pp. 463-476 ◽  
Author(s):  
H. J. Ryser
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Critical Survey ◽  
Combinatorial Properties ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Sum Vector

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class1.1consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).

On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

1999 ◽  
Vol 36 (1) ◽  
pp. 78-85 ◽  
Author(s):  
M. S. Sgibnev
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Stationary Distributions ◽  
Necessary And Sufficient ◽  
Submultiplicative Function ◽  
Independent Identically Distributed

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.

Unique Extension and Product Measures

1967 ◽  
Vol 19 ◽  
pp. 757-763 ◽  
Author(s):  
Norman Y. Luther
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Product Measure ◽  
Unique Extension ◽  
Product Measures ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Finite Measures

Following (2) we say that a measure μ on a ring is semifinite ifClearly every σ-finite measure is semifinite, but the converse fails.In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

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