Portmanteau theorem

The Portmanteau theorem , also called the Portmanteau theorem (alternative spelling also Portemanteau theorem or Portemanteau theorem ) is a theorem from the mathematical sub-areas of stochastics and measure theory . It lists equivalent conditions for the weak convergence of measures and their special case, the convergence in distribution of random variables . In some situations these conditions are easier to calculate than the definition of weak convergence. The sentence goes back to a work by Pawel Sergejewitsch Alexandrow from 1940, but is formulated in a wide variety of variants, different notation and generality, and is partly supplemented with independent mathematical sentences.

Formulations

The Portmanteau theorem essentially consists of three different types of statements:

1. The behavior of the sequences of (probability) measures on certain sets
2. The behavior in the formation of expected values ​​/ integration of certain function classes
3. Independent mathematical sentences that are included in the list.

These will vary depending on the author

• for finite measures , probability measures , sub-probability measures or as distributions of random variables
• on different basic quantities like , dem , Polish spaces or metric spaces ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$
• in the notation corresponding to the subject area (expected value vs. integral, vs. ) ${\displaystyle P}$ ${\displaystyle \mu }$

formulated.

Accordingly, many different formulations can be found in the literature. On the one hand, this article contains a formulation for the convergence in the distribution of real-valued random variables, which contains the most important statements for stochastics. The second formulation is a general, mass theoretical one. It can be adapted to special cases by applying appropriate restrictions.

Abbreviations and preliminary remarks

Important for the formulation of the theorem are the so- called - borderless sets , also called - continuity sets . If a Borel measure is on a Hausdorff space and Borel’s σ-algebra , then a set is called a -borderless set if its boundary is a -null set. It then applies ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle B}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$

${\displaystyle \mu (\partial B)=\mu ({\overline {B}}\setminus B^{\circ })=0}$,

where denotes the conclusion and the interior of the set . ${\displaystyle {\overline {B}}}$ ${\displaystyle B^{\circ }}$ ${\displaystyle B}$

Furthermore, be

• ${\displaystyle C_{b}^{g}(E)}$the space of uniformly continuous bounded functions ${\displaystyle E}$
• ${\displaystyle B(E)}$the space of limited functions on ${\displaystyle E}$
• ${\displaystyle \operatorname {Lip} (E)}$the space of Lipschitz continuous functions on ${\displaystyle E}$
• ${\displaystyle U_{f}}$ the set of all discontinuities of the function ${\displaystyle f}$

Formulation for distribution convergence of real random variables

Let be real-valued random variables. Then are equivalent: ${\displaystyle X,X_{1},X_{2},\dotsc }$

1. They converge in distribution to ${\displaystyle X_{n}}$ ${\displaystyle X}$
2. The distribution functions converge at every point of continuity from pointwise to ( Helly-Bray's theorem ). ${\displaystyle F_{X_{n}}}$ ${\displaystyle F_{X}}$ ${\displaystyle F_{X}}$
3. The characteristic functions converge pointwise to ( Lévy's continuity theorem ) ${\displaystyle \varphi _{X_{n}}}$ ${\displaystyle \varphi _{X}}$
4. It applies to everyone : ${\displaystyle f\in C_{b}^{g}(X)}$

   ${\displaystyle \lim _{n\to \infty }\operatorname {E} (f\circ X_{n})=\operatorname {E} (f\circ X)}$.

5. It's for everyone - rimless quantities. ${\displaystyle \lim _{n\to \infty }\mathbb {P} (X_{n}\in C)=\mathbb {P} (X\in C)}$ ${\displaystyle P_{X}}$
6. The following applies to all open quantities ${\displaystyle G}$

   ${\displaystyle \liminf _{n\to \infty }\mathbb {P} (X_{n}\in G)\geq \mathbb {P} (X\in G)}$.

7. For all closed sets applies ${\displaystyle A}$

   ${\displaystyle \limsup _{n\to \infty }\mathbb {P} (X_{n}\in A)\leq \mathbb {P} (X\in A)}$.

Dimension theory formulation

A metric space and the corresponding Borel σ-algebra are given . The following statements are equivalent for finite dimensions in the measuring space : ${\displaystyle (E,d)}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \mu ,\mu _{n}}$ ${\displaystyle (E,{\mathcal {B}})}$

• They converge weakly against ${\displaystyle \mu _{n}}$ ${\displaystyle \mu }$
• For all true ${\displaystyle f\in C_{b}^{g}(E)}$

  ${\displaystyle \lim _{n\to \infty }\int _{E}f\mathrm {d} \mu _{n}=\int _{E}f\mathrm {d} \mu }$

• For all true ${\displaystyle f\in B(E)\cap \operatorname {Lip} (E)}$

  ${\displaystyle \lim _{n\to \infty }\int _{E}f\mathrm {d} \mu _{n}=\int _{E}f\mathrm {d} \mu }$

• For all measurable with applies ${\displaystyle f\in B(E)}$ ${\displaystyle \mu (U_{f})=0}$

  ${\displaystyle \lim _{n\to \infty }\int _{E}f\mathrm {d} \mu _{n}=\int _{E}f\mathrm {d} \mu }$

• For each -randlose amount applies ${\displaystyle \mu }$ ${\displaystyle R\in {\mathcal {B}}}$

  ${\displaystyle \lim _{n\to \infty }\mu _{n}(R)=\mu (R)}$

• It is and for every open set is ${\displaystyle \lim _{n\to \infty }\mu _{n}(E)=\mu (E)}$ ${\displaystyle U}$

  ${\displaystyle \liminf _{n\to \infty }\mu _{n}(U)\geq \mu (U)}$.

• It is and for each closed set is ${\displaystyle \lim _{n\to \infty }\mu _{n}(E)=\mu (E)}$ ${\displaystyle A}$

  ${\displaystyle \limsup _{n\to \infty }\mu _{n}(A)\leq \mu (A)}$.

If also local compact and Polish , the list can be expanded to include the following two statements: ${\displaystyle (E,d)}$

• They vaguely converge on and ${\displaystyle \mu _{n}}$ ${\displaystyle \mu }$ ${\displaystyle \lim _{n\to \infty }\mu _{n}(E)=\mu (E)}$
• They vaguely converge on and ${\displaystyle \mu _{n}}$ ${\displaystyle \mu }$ ${\displaystyle \limsup _{n\to \infty }\mu _{n}(E)\leq \mu (E)}$

For finite dimensions, the following also applies: ${\displaystyle \mathbb {R} }$

• A sequence of finite measures on converges weakly to a measure if and only if a real sequence exists, so that the sequence of distribution functions (in the sense of measure theory) converges weakly on the distribution function of ( Helly-Bray theorem ). ${\displaystyle \mathbb {R} }$ ${\displaystyle \mu }$ ${\displaystyle (c_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (F_{n}-c_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$

Further formulations

There are other equivalent formulations for weak convergence. Sometimes there are other separating families (differentiable functions, restriction of properties by validity with the exception of a zero set, etc.). Not all of them are listed here.

There are also equivalent formulations of weak convergence, which are usually not included in the theorem. This includes, for example, the metrization of the corresponding topology using the Prokhorov metric or tightness criteria for the sequence of probability measures.

swell

1. ↑ Kusolitsch: Measure and probability theory. 2009, p. 290.
2. ^ RM Dudley: Real analysis and probability. Cambridge University Press, Cambridge 2002, ISBN 0-521-00754-2 , p. 433.

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , 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 , doi : 10.1007 / 978-3-642-36018-3 .
• Norbert Kusolitsch: Measure and probability theory. An introduction . 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .
• Patrick Billingsley: Convergence of probability measures. Wiley, New York 1999, ISBN 0-471-19745-9 .