Lothar Rogge

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.

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.

Compactness and domination

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

It is well known that each dominated family of probability measures is compact in the sense of Pitcher. In this paper the converse direction is investigated. It is shown that a family of probability measures on a-field is dominated if it is compact with respect to each sub- -field. If is only compact with respect to the basic-field settheoretical assumptions are needed to obtain domination.

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.

Filter und Ultrafilter

Wie schon in der Einleitung erwahnt, war es seit Leibniz ein Ziel der Mathematik, einen Erweiterungskorper *ℝ des Korpers ℝ der reellen Zahlen zu finden, der nicht-triviale infinitesimale Elemente enthalt und den man daruber hinaus fur die Analysis fruchtbringend verwenden kann. Die Erweiterung von ℝ zu *ℝ verlauft formal ahnlich wie die Erweiterung von ℚ zu ℝ. Die Konstruktion von ℝ aus ℚ kann in der Weise durchgefuhrt werden, das man in einer Klasse von ℚ-wertigen Folgen eine Aquivalenzrelation erklart. Die zugehorigen Aquivalenzklassen ergeben dann ℝ. Die Konstruktion von *ℝ aus ℝ erfolgt entsprechend mit Hilfe einer geeigneten Aquivalenzrelation in der Klasse aller ℝ-Wertigen Folgen; die Aquivalenzrelation wird dabei mit Hilfe eines Ultrafilters uber ℕ eingefuhrt. Die zugehorigen Aquivalenzklassen werden *ℝ liefern.

Ŝ-kompakte Nichtstandard-Einbettungen und die Standardteil-Abbildung

Ultrafilter sind grundlegend fur die Konstruktion von Nichtstandard-Modellen; sie werden sowohl bei der speziellen Konstruktion in § 3 als auch bei den allgemeinen Konstruktionen des § 36 verwandt. In diesem Paragraphen werden die Begriffe und Ergebnisse aus der inzwischen recht weit entwickelten Filtertheorie bereitgestellt, die fur diese Konstruktionen benotigt werden. Dies sind lediglich die Begriffe Filter und Ultrafilter, sowie zwei Ergebnisse uber Existenz und Charakterisierung von Ultrafiltern. Wer uber dieses elementare Rustzeug der Filtertheorie schon verfugt, kann diesen Paragraphen uberspringen.