The asymptotic validity of invariant procedures for the repeated measures model and multivariate linear model

Fairly general sufficient conditions are given to guarantee that invariant tests about means in the multivariate linear model and the repeated measures model have the correct asymptotic size when the normal assumption under which the tests are derived is relaxed. These conditions are the same as Huber's condition which guarantees asymptotic validity of the size of the F-test for the univariate linear model.     

On the loss of information due to fuzziness in experimental observations

The absence of exactness in the observation of the outcomes of a random experiment always entails a loss of information about the experimental distribution. This intuitive assertion will be formally proved in this paper by using a mathematical model involving the notions of fuzzy information and fuzzy information system (as intended by Tanaka, Okuda and Asai) and Zadeh's probabilistic definition. On the basis of this model we are first going to consider a family of measures of information enclosing some well-known measures, such as those defined by Kagan, Kullback-Leibler and Matusita, and then to establish methods for removing the loss of information due to fuzziness by increasing suitably the number of experimental observations.     

Non-anticipative representations of Banach space valued gaussian processes with respect to brownian motion

Necessary and sufficient conditions are found for the equivalence of the measures associated with (i) a Banach space valued Gaussian process, with mean 0, and (ii) a Bach space valued Brownian motion. The notion of a non-anticipative representation of (i) with respect to (ii) is defined and in the case of equivalence of the measures it is shown that such a representation exists and has an explicit stochastic integral form which is invertible. Theorems of Ershov on absolute continuity of measures associated with diffusion processes are extended to Banach space. Applications to infinite-dimensional filtering are considered.     

Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

On low order moments of the homodyned-K distribution

Fractional low order moments have been reported as beneficial for sampling computations using the K distribution. However, it has been recently pointed out that this it not the case for the homodyned-K distribution for a tissue discrimination problem. In this paper we show that such an statement is not fully justified. To that end, we follow a standard pattern recognition procedure both to determine class separability measures and to classify data with several classifiers. We conclude that the optimum order of the moments is intimately linked to the specific statistical properties of the tissues to be discriminated. Some ideas on how to choose the optimum order are discussed.     

Coin-tossing measures and their Fourier transforms

A coin-tossing measure μ on [0,1] is a probability measure satisfying

     

Lacunary Measures and Self-similar Probability Measuresin Function Spaces

The paper deals with multifractal quantities for some types of Radon measures, especially self-similar probability measures, and their relations to Besov spaces.     

Potential Analysis on Carnot Groups，Part II：Relationship between Hausdorff Measures and Capacities

Abstract In this paper, we establish the relationship between Hausdorff measures and Bessel capacities on any nilpotent stratified Lie group (i. e., Carnot group). In particular, as a special corollary of our much more general results, we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of . Given any set E ⊂ , B _α,p (E) = 0 implies ℋ ^{Q−αp+ ε}(E) = 0 for all ε > 0. On the other hand, ℋ ^Q−αp (E) < ∞ implies B _α,p (E) = 0. Consequently, given any set E ⊂ of Hausdorff dimension Q − d, where 0 < d < Q, B _α,p (E) = 0 holds if and only if αp ≤ d. A version of O. Frostman’s theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4). Research supported partly by the U. S. National Science Foundation Grant No. DMS99–70352     

BERNOULLI CONVOLUTIONS ASSOCIATED WITH CERTAIN NON——PISOT NUMBERS

The Bernoulli convolution Vλ measure is shown to be absolutely continuous with L^2 density for almost all 1/2＜λ＜1, and singular if λ^-1 is a Pisot number. It is an open question whether the Pisot type Bernoulli conuolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convo-lutions Vλ such that their density functions, if they exist, are not L^2. We also construct other Bernolulli convo-lutions whose density functions if they exist, behave rather badly.