Linear Inequalities and Polyhedra

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

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On the distribution of integer random variables related by two linear inequalities: I

Mathematical Notes ◽

10.1134/s0001434608030255 ◽

2008 ◽

Vol 83 (3-4) ◽

pp. 512-529 ◽

Cited By ~ 3

Author(s):

V. P. Maslov ◽

V. E. Nazaikinskii

Keyword(s):

Random Variables ◽

Linear Inequalities

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THE “KVAZAR” PACKAGE FOR PATTERN RECOGNITION AND ITS APPLICATIONS

International Journal of Software Engineering and Knowledge Engineering ◽

10.1142/s0218194093000215 ◽

1993 ◽

Vol 03 (04) ◽

pp. 439-444 ◽

Cited By ~ 2

Author(s):

VLADIMIR S. KAZANTSEV

Keyword(s):

Experimental Data ◽

Pattern Recognition ◽

Data Analysis ◽

Linear Inequalities ◽

Processing Data ◽

Experimental Data Analysis ◽

Ibm Pc

The package of applied programs named KVAZAR has been elaborated to be used for classification, diagnostic, predicative, experimental data analysis problems. The package may be used in medicine, biology, geology, economics, engineering and some other problems. The algorithmical base of the package is the method of pattern recognition, based on the linear inequalities and committee constructions. Other algorithms are used too. The package KVAZAR is intended to be used with IBM PC AT/XT. The range of processing data is bounded by 40,000 numbers.

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