Dieter Landers

The Hahn-Vitali-Saks and the Uniform Boundedeess Theorem in topological groups

It is shown in this paper that the Theorem of Hahn-Vitali-Saks (Theorem 4) and the Uniform Boundedness Theorem (Theorem 5) hold true for measures with values in a topological group. The proofs given here for these theorems seem to the authors to be more direct than the usual proofs for real valued measures.

Connectedness properties of the range of vector and semimeasures

It is shown that the range of a non-atomic semimeasure as well as the range of a non-atomic measure defined on a δ-ring with values in a Banach space or even in an abelian topological Hausdorff group is arcwise connected.

On projections and monotony in Lp -spaces

In this paper we investigate the connection between the range of nearest point projections in Lp -spaces and monotony properties of the projection operator. We show e.g. that a nearest point projection onto a closed convex subset of an Lp -space (1<p<∞) is monotone if and only if the closed convex range is a lattice. If the range is closed linear instead of closed convex then it turns out that positivity of the projection operator implies monotony, although the projection is in general not a linear operator. We can apply these results to a lot of known cases and to a case, in which the monotony of the projection operator was unknown up to now.

Uniform normal approximation orders for families of dominated measures

Let (Q, JZ/, P) be a probability space and 1 0: 1x1 6 (’ P-a.e.}. Let X,, E SC: (9, .d, P, Iw”), n E N, be a sequence of independent and identically distributed (i.i.d.) random vectors with positive definite covariance matrix V. Put S,T = (l/J%) V-“‘(C;=, (X,.P[X,])), where P[X,] = j X, dP. Let G

EQUICONTINUITY AND CONVERGENCE OF MEASURES

= cr(X, ,..., X,,) be the a-field generated by X, ,..., X,. If cp E 9, (Q, .d. P, IF!), let

Nonstandard methods for families of -smooth probability measures

It is shown in this paper that each family of measures with values in an abelian topological group which is equicontinuous on a ring is equicontinuous on the generated σ-ring. A family of measures is equicontinuous iff the corresponding family of “semivariations” is equicontinuous. It is furthermore shown that a family of measures which is equicontinuous and Cauchy convergent on a ring is Cauchy convergent on the generated σ-ring. A family of measures which is Cauchy convergent for all countable sums of elements of a ring is Cauchy convergent on the generated σ-ring.

Joint convergence of conditional expectations

For families of T-smooth probability measures we give nonstandard characterizations of uniform T-smoothness, uniform tightness, uniform pretightness, and relative compactness in the weak topology. We apply these characterizations to obtain two important theorems of probability theory: A theorem of Tops0e and a theorem of Prohorov.

Nonstandard Characterization of Convergence in Law for D[0,1]-Valued Random Variables

Let Pn, n∈IN∪{0}, be probability measures on a-fieldA; fn, n∈IN∪{0}, be a family of uniformly boundedA-measurable functions andAn, n∈IN, be a sequence of sub--fields ofA, increasing or decreasing to the-fieldAo. It is shown in this paper that the conditional expectations converge in Po-measure to with k, n, m → ∞, if Pn|A, n∈IN, converges uniformly to Pn|A and fn, n∈IN, converges in Po-measure to fo.

Nonstandrad Characterization for a General Invariance Principle

We prove for random variables with values in the space D[0,1] of cadlag functions - endowed with the supremum metric - that convergence in law is equivalent to nonstandard constructions of internal S-cadlag processes, which represent up to an infinitesimal error the limit process. It is not required that the limit process is concentrated on the space C[0,1], so that the theory is applicable to a wider class of limit processes as e.g. to Poisson processes or Gaussian processes. If we consider in D[0,1] the Skorokhod metric - instead of the supremum metric - we obtain a corresponding equivalence to constructions of internal processes with S-separated jumps. We apply these results to functional central limit theorems.

Der Erweiterungskörper *ℝ von ℝ

In this paper it is shown that every invariance principle of probability theory is equivalent to a nonstandard construction of internal S-continuous processes, which all represent — up to an infinitesimal error — the limit process. This can be applied e.g. to obtain Anderson’s nonstandard construction of a Brownian motion on a hyperfinite set.