Prokhorov theorem

The set of Prokhorov is a sentence from the measure theory , a branch of mathematics that the investigation focuses on abstracted volume terms. These form the basis for stochastics and integration theory . Sometimes there is also the spelling Prohorov's sentence or Prokhorov's sentence , which was adopted from English . The set provides criteria under which sets of measures are relatively sequentially compact with respect to weak convergence . Thus, sequences of measures from such sets always have a weakly convergent subsequence . The sentence is named after Yuri Wassiljewitsch Prokhorov , who published it in 1956.

statement

Given a metric space and a family of finite measures on the associated Borel σ-algebra . Then: ${\ displaystyle (X, d)}$ ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle {\ mathcal {B}} (X)}$

If the family is tight and restricted, it is also relatively compact in terms of consequences in terms of weak convergence . ${\ displaystyle {\ mathcal {M}}}$

If there is a Polish area , the reverse also applies. It follows from this that under these conditions it is tight and restricted if and only if weak is relatively compact in consequence. ${\ displaystyle (X, d)}$ ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle {\ mathcal {M}}}$

Here is a set of measures is limited, if the amount of total variation norms in limited is. ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle \ {\ | \ mu \ | \; | \; \ mu \ in {\ mathcal {M}} \}}$ ${\ displaystyle \ mathbb {R}}$

variants

In probability theory, the theorem is sometimes formulated only for sets of probability measures ; the restriction condition is then dispensed with, since it is always fulfilled.

A special case of this for probability measures on the real numbers is to formulate Prokhorov's theorem only for distribution functions in the sense of probability theory and then to make the connection to weak convergence via the Helly-Bray theorem . A family of distribution functions is a tight family of distribution functions , if at any one exists, so ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle {\ mathcal {F}} = (F_ {i}) _ {i \ in I}}$ ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle C \ in \ mathbb {R}}$

{\ displaystyle \ inf _ {i \ in I} \ {F_ {i} (C) -F_ {i} (- C) \}> 1- \ varepsilon}

is. Since is Polish, Prokhorov's theorem reads that a family of distribution functions is tight if and only if every sequence in this family has a weakly convergent subsequence of distribution functions . ${\ displaystyle \ mathbb {R}}$

Individual evidence

↑ Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 296 , doi : 10.1007 / 978-3-642-45387-8 .

literature

Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 392-398 , doi : 10.1007 / 978-3-540-89728-6 .
Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 265-275 , doi : 10.1007 / 978-3-642-36018-3 .

[1] Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 296 , doi : 10.1007 / 978-3-642-45387-8 .