We present new sufficient conditions for the asymptotic stability of Markov operators acting on the space of signed measures. Our results are based on two principles. The first one is the LaSalle invariance principle used in the theory of dynamical systems. The second is related to the Kantorovich-Rubinstein theorems concerning the properties of probability metrics. These criteria are applied to stochastically perturbed dynamical systems, a Poisson driven stochastic differential equation and a mathematical model of the cell cycle. Moreover, we discuss the problem of the asymptotic stability of solutions of a generalized version of the Tjon-Wu equation.
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A new sufficient condition for the asymptotic stability of a locally Lipschitzian Markov semigroup acting on the space of signed measures $𝓜 _{sig}$ is proved. This criterion is applied to the semigroup of Markov operators generated by a Poisson driven stochastic differential equation.
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A new version of the maximum principle is presented. The classical Kantorovich-Rubinstein principle gives necessary conditions for the maxima of a linear functional acting on the space of Lipschitzian functions. The maximum value of this functional defines the Hutchinson metric on the space of probability measures. We show an analogous result for the Fortet-Mourier metric. This principle is then applied in the stability theory of Markov-Feller semigroups.
1. Introduction Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7]. We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument $x(t,ω) = h(t,ω) + ∫^{t+δ(t)}₀ k(t,τ,ω)f(τ,x_τ(ω))dτ$, (t,ω) ∈ R⁺ × Ω, (1) where (i) (Ω,A,P) is a complete probability space, (ii) x = x(t,ω) denotes an unknown random function defined for t ∈ R⁺ and ω ∈ Ω, (iii) δ is a nonnegative function from R⁺ into R⁺, (iv) xₜ(ω) denotes the restriction of the function x(t,ω) to the interval [0,t+δ(t)], t>0, with x₀(ω) = x(0,ω) ∈ L²(Ω,A,P).
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We present a new necessary and sufficient condition for the asymptotic stability of Markov operators acting on the space of signed measures. The proof is based on some special properties of the total variation norm. Our method allows us to consider the Tjon-Wu equation in a linear form. More precisely a new proof of the asymptotic stability of a stationary solution of the Tjon-Wu equation is given.
W artykule omówiono problem klasyfikacji dla dwóch klas w przypadku przyjęcia założenia, że rozkłady cech w klasach są wielowymiarowymi rozkładami normalnymi. Problem rozwiązano za pomocą empirycznego klasyfikatora gaussowskiego i wybranych estymatorów nieznanych parametrów wielowymiarowego rozkładu normalnego. Uwzględnione zostały następujące estymatory: MLE (the maximum likelihood estimator - estymator największej wiarogodności), KZE (Kulawik-Zontek estimator) i MCDE (the minimum covariance determinant estimator). Klasyfikatory oparte o MLE i KZE zostały porównane w przypadku przykładu empirycznego (mała próba). W przypadku dużych prób porównane zostały klasyfikatory oparte o trzy wspomniane estymatory.
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In the article, a classification problem with two distributed classes is considered. The problem is solving using empirical discriminant functions for Gaussian classifier and estimators for unknown parameters of multivariate normal distribution. The three etimators, maximum likelihood estimator, Kulawik-Zontek estimator and minimum covariance determinant estimator, are compared in two different empirical examples (small size sample and large size sample).