The kurzweil construction of an integral in ordered spaces

This paper generalizes the results of papers which deal with the Kurzweil-Henstock construction of an integral in ordered spaces. The definition is given and some limit theorems for the integral of ordered group valued functions defined on a Hausdorff compact topological space T with respect to an ordered group valued measure are proved in this paper.     

A Note on the Comedian for Elliptical Distributions

The comedianCOM(X, Y) of random variablesX,Yis a median based robust alternative to the covariance ofXofY. For the bivariate normal case it is known thatCOM(X, Y), standardized by the median absolute deviations ofXandY, is a symmetric, strictly increasing and continuous function of the correlation coefficientρwith range [−1, 1] and can therefore serve as a robust alternative toρ. We show that this result, which is not true in general, extends to elliptical distributions even in the case where moments ofX,Ydo not exist.     

Discrete Spectrum of Nonstationary Stochastic Processes on Groups

Vector-valued, asymptotically stationary stochastic processes on -compact locally compact abelian groups are studied. For such processes, we introduce a stationary spectral measure and show that it is discrete if and only if the asymptotically stationary covariance function is almost periodic. Using an almost periodic Fourier transform we recover the discrete part of the spectral measure and construct a natural, consistent estimator for the latter from samples of the process.     

Hausdorff Measures and Capacities for Increase Times of Stable Processes

Let X be a real-valued stable process. It is known that X possesses increase times iff (X ₁ > 0) > ([1]). In that case, we specify the exact Hausdorff function of the set of increase times, and characterize the subsets of [0,+) which contain increase times in terms of their capacity.     

A generalization of Lyapunov's convexity theorem with applications in optimal stopping

Lyapunov proved that the range of finite measures defined on the same -algebra is compact, and if each measure also is atomless, then the range is convex. Although both conclusions may fail for measures on different -algebras of the same set, they do hold if the -algebras are nested, which is exactly the setting of classical optimal stopping theory.

     

An estimate on the distortion of the logarithmic capacity

Let be a Fuchsian group. We show that the existence of a set on with no -equivalent points and positive logarithmic capacity does not imply that is of convergence type.

     

Carleson conditions for asymptotic weights

The doubling and conditions are characterized in terms of convolution with rapidly decreasing kernels. The Carleson-measure criterion for of Fefferman, Kenig, and Pipher is extended to the case when all bounds become optimally small in the asymptotic limit.

     

Characterizations of weakly compact operators on

Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

     

Doob-meyer decomposition for set-indexed submartingales

Set-indexed martingales and submartingales are defined and studied. The admissible function of a submartingale is defined and some class (D) conditions are given which allow the extension of the function to a -additive measure on the predictable -algebra. Then, we prove a Doob-Meyer decomposition: A set-indexed submartingale can be decomposed into the sum of a weak martingale and an increasing process. A hypothesis of predictability ensures the uniqueness of this decomposition. An explicit construction of the increasing process associated with a submartingale is given. Finally, some remarks, about quasimartingales are discussed.Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Dept. of Math. University of Ottawa, Ottawa, Ontario, Canada, K1N 6N5.The third author wishes to thank Professor Ivanoff and Dr. Dozzi for their kind hospitality. Dept. of Math. & Comp. Sci. Bar-Ilan University, Ramat-Gan 52900, Israel.     

Probabilistic Adaptive Width of a Multivariate Sobolev Space Equipped with a Gaussian Measure

In this paper, we determine the asymptotic values of the probabilistic adaptive widths of the space of multivariate functions with bounded mixed derivative (MW₂^r(T^d),μ) relative to the manifold (Y^N,ν) in the L_q(T^d)-norm, 1 < q ≤ 2, where μ and ν are two given Gaussian measures.