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Journal of Applied Analysis Vol. 14, No. 1 (2008), pp. 13–26 DERIVATIVES OF MARKOV KERNELS AND THEIR JORDAN DECOMPOSITION B. HEIDERGOTT, A. HORDIJK and H. WEISSHAUPT∗ Received April 5, 2005 and, in revised form, September 6, 2006 Abstract. We study a particular class of transition kernels that stems from differentiating Markov kernels in the weak sense. Sufficient con- ditions are established for this type of kernels to admit a Jordan-type decomposition. The decomposition is explicitly constructed. 1. Introduction Let (Pϑ)ϑ∈Θ be a parametric family of Markov

References [1] Devroye, L. (1987). A Course in Density Estimation. Birkhäuser, Boston MA. [2] Durante, F. and C. Sempi (2016). Principles of Copula Theory. CRC Press, Boca Raton FL. [3] Fernández-Sánchez, J. and W. Trutschnig (2015). Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and iterated function systems. J. Theoret. Probab. 28(4), 1311-1336. [4] Fernández-Sánchez, J. and W. Trutschnig (2016). Some members of the class of (quasi-)copulas with given diagonal from the Markov kernel

Abstract

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p f, a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets.

Reliability Modeling of Fault Tolerant Control Systems

This paper proposes a novel approach to reliability evaluation for active Fault Tolerant Control Systems (FTCSs). By introducing a reliability index based on the control performance and hard deadline, a semi-Markov process model is proposed to describe system operation for reliability evaluation. The degraded performance of FTCSs in the presence of imperfect Fault Detection and Isolation (FDI) is reflected by semi-Markov states. The semi-Markov kernel, the key parameter of the process, is determined by four probabilistic parameters based on the Markovian model of FTCSs. Computed from the transition probabilities of the semi-Markov process, the reliability index incorporates control objectives, hard deadline, and the effects of imperfect FDI, a suitable quantitative measure of the overall performance.

Statistics & Decisions 15, 319 - 347 (1997) © R. Oldenbourg Verlag, München 1997 ON SUFFICIENCY AND BLACKWELL SUFFICIENCY J. Hille Received: Revised version: December 20, 1996 A b s t r a c t Let (Χ,Λ,Ρ) be a stat ist ical experiment . We show t h a t a sub-a-field Β of A is weakly Blackwell sufficient if and only if it contains a classically sufficient σ-field. This is done in the more general se tup of sufficiency of weak Markov kernels. From this we deduce the following converse of a well-known result of Bahadur [1]: If for every decision funct ion δ

], Lauritzen et al. [20], Neapolitan [22], Kruse et al. [18] and Hajek et al. [11]. Such a CPN is a directed graph G := (V,E) with E C V χ V and (u, υ) φ Ε for (v, u) £ Ε and a family Ρ := (Pv : ν € V) of Markov kernels Pv : S(Pa(v)) χ &υ -> [0,1] with ((xu : u e Ρα(υ), Β) ·-» Pv((xu : u <Ξ Ρα(ν); Β) where Pa(v) := {u e V : (u, v) 6 E} is the set of the parents of v in G, S(U) := fingt; Sufor 0 ^ U CV is the product of the state spaces of the family of random variables (Xu : u 6 U], and S(0) x &v := &v. Every node υ e V represents the random variable Xv with the state space

first step toward the inclusion of the duration time dependence into the model. They assume that the future evolution of the system depends on the history only through the current state and the duration time in the current state. Many efforts have been spent to study statistical inference for these semi–Markov models. Indeed, Lagakos et al. (1978) presented nonpara- metric maximum likelihood estimation for the semi–Markov kernel, proposing a plug-in estimator. Their approach allows an arbitrary number of states as well as right censored observations. In Gill (1980

–1336. [12] J. Fernández-Sánchez and W. Trutschnig (2016). Some members of the class of (quasi-) copulas with given diagonal from the Markov kernel perspective. Commun. Stat. Theor. Meth. 45(5), 1508–1526. [13] G.A. Fredricks, R.B. Nelsen and J.A. Rodríguez-Lallena (2005). Copulaswith fractal supports. Insur.Math. Econ. 37(1), 42–48. [14] C. Genest, J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and C. Sempi (1999). A characterization of quasi-copulas. J. Multivariate Anal. 69(2), 193–205. [15] M. Grabisch, J.-L. Marichal, R. Mesiar, and E. Pap (2009). Aggregation

and Cores for Fuzzy Games with Infinitely Many Players, International Journal of Game Theory 16 (1987), 43-68. [7] BUTNARIU, D.-KLEMENT, E. P. : Triangular Norm-Based Measures and their Markov Kernel Representation, Journal of Mathematical Analysis and Applications 162 (1991), 111-143. [8] De BAETS, B.-MESIAR, M. : Triangular Norms on Product Lattices, Fuzzy Sets and Systems 104 (1999), 61-75. [9] DVORETZKY, A.-WALD, A.-WOLFOWITZ, J. : Relations Among Certain Ranges of Vector Measures, Pacific Journal of Mathematics 1 (1951), 59-74. [10] FRANK, M. D. : On the

'learning rule for Boltzmann machines', cf. [18], Section 15.3, and [1]. The adaptive algorithm below makes these ideas precise in an elegant fashion. We restrict ourselves to the Gibbs sampler with deterministic visiting scheme. Let prt denote the projection X —> Gt, χ xt. For a Gibbs field 77 let n(ys\xt, t φ s) = Π(ρτ3 = ys\prt = xt, t φ s) denote the single-site conditional probabilities. Fix a visiting scheme, i.e. an enumeration {ä!, . . . , S|s|} of S and define a Markov kernel Ρ on X by |S| P(x,y) = Π77^'!^' J < > *)> x>y e X (4) i=l Ρ is said to govern