Difference between revisions of "Dynkin system"
m |
m |
||
| Line 5: | Line 5: | ||
===[[Dynkin system/Definition 2|Second Definition]]=== | ===[[Dynkin system/Definition 2|Second Definition]]=== | ||
{{:Dynkin system/Definition 2}} | {{:Dynkin system/Definition 2}} | ||
| − | + | ==Proof of equivalence of definitions== | |
| + | {{Todo|Do this}} | ||
==Immediate results== | ==Immediate results== | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
Revision as of 23:27, 2 August 2015
Note: a Dynkin system is also called a "[ilmath]d[/ilmath]-system"[1] and the page d-system just redirects here.
Contents
Definition
First Definition
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]Given a set [ilmath]X[/ilmath] and a family of subsets of [ilmath]X[/ilmath], which we shall denote [ilmath]\mathcal{D}\subseteq\mathcal{P}(X)[/ilmath] is a Dynkin system[2] if:
- [ilmath]X\in\mathcal{D} [/ilmath]
- For any [ilmath]D\in\mathcal{D} [/ilmath] we have [ilmath]D^c\in\mathcal{D} [/ilmath]
- For any [ilmath](D_n)_{n=1}^\infty\subseteq\mathcal{D}[/ilmath] is a sequence of pairwise disjoint sets we have [ilmath]\udot_{n=1}^\infty D_n\in\mathcal{D}[/ilmath]
Second Definition
Given a set [ilmath]X[/ilmath] and a family of subsets of [ilmath]X[/ilmath] we denote [ilmath]\mathcal{D}\subseteq\mathcal{P}(X)[/ilmath] is a Dynkin system[3] on [ilmath]X[/ilmath] if:
- [ilmath]X\in\mathcal{D} [/ilmath]
- [ilmath]\forall A,B\in\mathcal{D}[B\subseteq A\implies A-B\in\mathcal{D}][/ilmath]
- Given a sequence [ilmath](A_n)_{n=1}^\infty\subseteq\mathcal{D}[/ilmath] that is increasing[Note 1] and has [ilmath]\lim_{n\rightarrow\infty}(A_n)=A[/ilmath] we have [ilmath]A\in\mathcal{D}[/ilmath]
Proof of equivalence of definitions
TODO: Do this
Immediate results
- [ilmath]\emptyset\in\mathcal{D} [/ilmath]
Proof:
- As [ilmath]\mathcal{D} [/ilmath] is closed under complements and [ilmath]X\in\mathcal{D} [/ilmath] by definition, [ilmath]X^c\in\mathcal{D} [/ilmath]
- [ilmath]X^c=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{D} [/ilmath]
This completes the proof.
See also
- Dynkin system generated by
- Types of set algebras
- [ilmath]p[/ilmath]-system
- Conditions for a [ilmath]d[/ilmath]-system to be a [ilmath]\sigma[/ilmath]-algebra
Notes
- ↑ Recall this means [ilmath]A_{n}\subseteq A_{n+1} [/ilmath]
References
- ↑ Probability and Stochastics - Erhan Cinlar
- ↑ Measures, Integrals and Martingales - René L. Schilling
- ↑ Probability and Stochastics - Erhan Cinlar
