Some remarks on the relation between the Harker–Kasper inequality and the Okaya–Nitta linear inequalities

1952 ◽  
Vol 5 (5) ◽  
pp. 697-697 ◽  
Author(s):  
K. Sakurai
Keyword(s):  
Linear Inequalities

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

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 $\boldsymbol{\pi}(N)$ .

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

2008 ◽  
Vol 83 (3-4) ◽  
pp. 512-529 ◽  
Author(s):  
V. P. Maslov ◽  
V. E. Nazaikinskii
Keyword(s):  
Random Variables ◽  
Linear Inequalities

THE “KVAZAR” PACKAGE FOR PATTERN RECOGNITION AND ITS APPLICATIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 439-444 ◽  
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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