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In this article we evaluate P ( X < Ω) and P ( X > Ω), when X has a matrix variate Kummer-beta or Kummer-gamma distribution and Ω is a non-random positive definite matrix. These results are obtained in series involving invariant polynomials of matrix arguments.
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We present the random representations for the Navier-Stokes vorticity equations for an incompressible fluid in a smooth manifold with smooth boundary and reflecting boundary conditions for the vorticity. We specialize our constructions to R n− 1 × R + . We extend these constructions to give the random representations for the kinematic dynamo problem of magnetohydrodynamics. We carry out these integrations through the application of the methods of Stochastic Differential Geometry, i.e. the gauge theory of diffusion processes on smooth manifolds.
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This paper introduces a class of semilinear stochastic fractional differential equations of Volterra type. The existence and uniqueness of their solutions is proved and some basic properties of the solutions are studied. A simulation scheme is proposed which converges uniformly in mean square for a special, but important, case.
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In this article, we study a filtration of Furstenberg-Kifer type for Lyapunov exponents of a degenerate random dynamical system described by an implicit linear equation. An application of this result is to give a proof of the existence of bounded solutions.
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After many years of investigations in the Theory of Random Matrices , we can say today that a very important and advanced result occupies the central place in this theory: CENTral REsolvent Law (CENTRE-LAW) for the traces of analytic function of random matrices, proved in 1975, in [15, pp. 278–324]. In the present paper we continue to consider this important problem of Theory of Random Matrices (TRM) - the CENTRE-LAW for the resolvent’s trace of a certain empirical covariance matrix of dimension m n which is used in almost all known estimators of General Statistical Analysis(GSA). At the end of this paper the reader can find the literature concerning GSA: [1–46, GSA]. Here we follow the main procedures of REFORM method (REsolvent, FORmula and Martingale) and have shown as 30 years ago that Central Limit Theorem for the traces of analytic function of random matrices has an unbelievable property: it is asymptotically normal with convergence rate ( m n n ) −1/2 under G-condition m n n −1 <1, where n is the number of independent observations of a random vector with covariance matrix . We want to emphasize that all known publications concerning the problem of estimation of functions of many parameters deal only with improvements of estimators. See, for example, jackknife and bootstrap methods. Only in [1–46,GSA] it was for the first time, shown that there exist in this analysis consistent estimators of some functions under the G -condition. Therefore, we can develop mathematical statistics under G -condition without any new restrictions for observations and statistical models. In the following sections we present a review of the main steps of the proof of the main assertions about Central Resolvent Law for the traces of analytic function of random matrices. We describe very succinctly the main features of the proof of the CENTRE-law. As in the previous papers, we will focus mainly on the limit theorem for random determinants. The proof is quite similar to the one proved in [15, pp. 278–324].