Bertram M. Schreiber

A new factorization property of the selfdecomposable probability measures

We prove that the convolution of a selfdecomposable dis- tribution with its background driving law is again selfdecomposable if and only if the background driving law is s-selfdecomposable. We will refer to this as the factorization property of a selfdecomposable distribution; let L f denote the set of all these distributions. The algebraic structure and various characterizations of L f are studied. Some examples are discussed, the most interesting one being given by the Levy stochastic area integral. A nested family of subclasses L f,n � 0, (or a filtration) of the class L f is given.

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C0(G) × C0(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a KG2-Banach ∗-algebra, where KG is Grothendiecks universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ∗-representations of BM(G, H) are investigated.

Algebraic models for probability measures associated with stochastic processes

This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foias notion of algebraic models for probability measures. The main result is a general extension theorem of Kolmogorov type which can be summarized as follows: Let {(X, si¡, ¡i,), i e 1} be a directed family of probability measure spaces. Then there is an associated directed family of probability measure spaces {(G, &t, vt), i e 1} and a probability measure v on the »-algebra M generated by the 3)¡ such that (i) v(B) = v¡(B), Be &,, ie I, and (ii) for each iel the spaces (X, s/t, /it) and (G, &lt v¡) are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding ip: X—* G such that the induced measures vt on G are extendable to v, although the measures n¡ on X may not be extendable. Using the algebraic model formulation, the Kolmogorov extension property and the notion of a representation of a directed family of probability measure spaces are discussed.

Representation of certain infinitely divisible probability measures on Banach spaces

Subclasses L0 [superset or implies] L1 [superset or implies] ... [superset or implies] L[infinity] of the class L0 of self-decomposable probability measures on a Banach space are defined by means of certain stability conditions. Each of these classes is closed under translation, convolution and passage to weak limits. These subclasses are analogous to those defined earlier by K. Urbanik on the real line and studied in that context by him and by the authors. A representation is given for the characteristic functionals of the measures in each of these classes on conjugate Banach spaces. On a Hilbert space it is shown that L[infinity] is the smallest subclass of L0 with the closure properties above containing all the stable measures.

Discrete Spectrum of Nonstationary Stochastic Processes on Groups

Vector-valued, asymptotically stationary stochastic processes on σ-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.

Fourier transforms of measures from the classes U b ′ -2< b ≤- 1

Subclasses [beta](E), -2 [[integral operator](0,1) t dY(t[beta])] are investigated when E is a Hilbert space. As an application of the fact that [beta] is a continuous isomorphism, generators for [beta] are found as the images of compound Poisson distributions. Finally, the connection between the distributions [beta] and thermodynamic limits in the Ising model with zero external field is pointed out.

Approximation and extension of random functions

Stochastic versions of the extension theorems of Tietze and Dugundji are obtained, as well as an existence theorem for partitions of unity by random continuous functions. A form of the classical approximation theorem of Mergelyan valid for random holomorphic functions on random compact sets is presented. A similar approach yields versions of the approximation theorems of Runge, Arakelyan, and Vitushkin.

PROJECTIONS IN SPACES OF BIMEASURES

Let X and Y be metrizable compact spaces and /x and v be nonzero continuous measures on X and Y, respectively. Then there is no bounded operator from the space of bimeasures BM(X, Y) onto the closed subspace of BM(X, Y) generated by L (ft X v); in particular, if Xand Fare nondiscrete locally compact groups, then there is no bounded projection from BM(X, Y) onto the closed subspace of BM(X, Y) generated by Ll(X X Y). 0. Introduction and Statement of Results. Let X, Y and Z be locally compact Hausdorff spaces. The space of bounded, regular Borel measures on X is denoted by M(X). The tensor algebras V0(X, Y) and V0(X9 Y, Z) are the respective closures, C0(X)ec0(Y) and C0(X) O C0(Y) O C0(Z), in the greatest cross-norm (projective norm), of the tensor products of the indicated C0-spaces. The space BM(X9 Y) of bimeasures on X X Y constitutes the dual space of V0(X9 Y); the dual space of V0(X Y, Z) will be denoted by BM(X, Y9 Z) and its elements will be called trimeasures. Given a measure co, we consistently identify Ll(co) with the space of measures that are absolutely continuous with respect to C0(Y)* be the operator given by =u(f® g), / e C0(X)9 g G C0(Y). Thus S**:C0(X)** -» C 0(Y)***. For

An iterated logarithm law for families of Brownian paths

G C0(X)** and * G C0(Y)**, set

Asymptotically stationary processes on amenable groups

SummaryIf n is large, a plot of log n independent Brownian paths over [0, n] is nearly certain to give the appearance of a shaded region having square root boundaries.