Zufällige Menge

Eine zufällige Menge ist eine Menge, deren Charakteristika (z. B. Größe, Gestalt, Lage) auch vom Zufall abhängen, z. B. die raum-zeitliche Entwicklung einer Epidemie, oder eines Ölteppiches auf dem Ozean. Zufällige Mengen sind auch grundlegend für die stochastische Geometrie.

Definition

Eine zufällige Menge ist eine mengenwertige Zufallsvariable, d. h. eine messbare Abbildung von einem Wahrscheinlichkeitsraum in einen messbaren Raum . Häufig ist die Menge aller kompakten Teilmengen eines lokalkompakten separablen Hausdorff-Raumes und die von erzeugte Sigma-Algebra. Dann spricht man von einer zufälligen kompakten Menge, siehe z. B.[1]

Verteilung einer zufälligen kompakten Menge

Sei eine zufällige kompakte Menge. Die Verteilung von ist eindeutig festgelegt durch die Wahrscheinlichkeiten, mit denen beliebige 's aus "trifft" (sog. hit-probabilities), d. h.

ist eine vollständig alternierende Kapazität.

Erwartungswert einer zufälligen kompakten Menge

Sei eine zufällige kompakte Menge. Ihr Erwartungswert wird häufig Aumann-Erwartungswert genannt[2]. Er ist definiert als die Menge aller Erwartungswerte von Zufallsgrößen , die fast sicher in liegen, d. h.

.

Die werden auch Selektoren von genannt. Für ein zufälliges Intervall ergibt sich z. B.

.

Der Aumann-Erwartungswert ist linear bzgl. der Minkowski-Summe , d. h.

.

Literaturhinweise

  • Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York.
  • Molchanov, I. (2005) The Theory of Random Sets. Springer, New York.
  • Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields. John Wiley & Sons, Chichester, New York.

Einzelnachweise

  1. ↑ Matheron, G. (1975) Random Sets and Integral Geometry. J.Wiley & Sons, New York.
  2. ↑ Aumann.J.(1965). Integral of set valued functions. Journ.Math.Anal.Appl.12, 1-22.

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