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In [[mathematics]]{{spaced ndash}}specifically, in [[stochastic processes|stochastic analysis]]{{spaced ndash}}'''Doob's martingale convergence theorems''' are a collection of results on the long-time [[limit (mathematics)|limits]] of [[Martingale (probability theory)|supermartingale]]s, named after the American mathematician [[Joseph L. Doob]].


==Statement of the theorems==
In the following, (Ω,&nbsp;''F'',&nbsp;''F''<sub>∗</sub>,&nbsp;'''P'''), ''F''<sub>∗</sub>&nbsp;=&nbsp;(''F''<sub>''t''</sub>)<sub>''t''≥0</sub>, will be a [[filtered probability space]] and ''N''&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;&times;&nbsp;Ω&nbsp;→&nbsp;'''R''' will be a right-[[continuous function|continuous]] supermartingale with respect to the filtration ''F''<sub>∗</sub>;  in other words, for all 0&nbsp;≤&nbsp;''s''&nbsp;≤&nbsp;''t''&nbsp;&lt;&nbsp;+∞,


:<math>N_{s} \geq \mathbf{E} \big[ N_{t} \big| F_{s} \big].</math>
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===Doob's first martingale convergence theorem===
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Doob's first martingale convergence theorem provides a sufficient condition for the [[random variable]]s ''N''<sub>''t''</sub> to have a limit as ''t''&nbsp;→&nbsp;+∞ in a pointwise sense, i.e. for each ''ω'' in the [[sample space]] Ω individually.
 
For ''t''&nbsp;≥&nbsp;0, let ''N''<sub>''t''</sub><sup>&minus;</sup>&nbsp;=&nbsp;max(&minus;''N''<sub>''t''</sub>,&nbsp;0) and suppose that
 
:<math>\sup_{t > 0} \mathbf{E} \big[ N_{t}^{-} \big] < + \infty.</math>
 
Then the pointwise limit
 
:<math>N(\omega) = \lim_{t \to + \infty} N_{t} (\omega)</math>
 
exists finite for '''P'''-[[almost all]] ''ω''&nbsp;∈&nbsp;Ω.
 
===Doob's second martingale convergence theorem===
 
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any [[Lp space|''L''<sup>''p''</sup> space]]. In order to obtain convergence in ''L''<sup>1</sup> (i.e., [[convergence of random variables|convergence in mean]]), one requires uniform integrability of the random variables ''N''<sub>''t''</sub>. By [[Chebyshev's inequality]], convergence in ''L''<sup>1</sup> implies convergence in probability and convergence in distribution.
 
The following are equivalent:
 
* (''N''<sub>''t''</sub>)<sub>''t''&gt;0</sub> is [[uniformly integrable]], i.e.
 
::<math>\lim_{C \to \infty} \sup_{t > 0} \int_{\{ \omega \in \Omega | N_{t} (\omega) > C \}} \big| N_{t} (\omega) \big| \, \mathrm{d} \mathbf{P} (\omega) = 0;</math>
 
* there exists an [[Integral|integrable]] random variable ''N''&nbsp;∈&nbsp;''L''<sup>1</sup>(Ω,&nbsp;'''P''';&nbsp;'''R''') such that ''N''<sub>''t''</sub>&nbsp;→&nbsp;''N'' as ''t''&nbsp;→&nbsp;+∞ both '''P'''-[[almost surely]] and in ''L''<sup>1</sup>(Ω,&nbsp;'''P''';&nbsp;'''R'''), i.e.
 
::<math>\mathbf{E} \big[ \big| N_{t} - N \big| \big] = \int_{\Omega} \big| N_{t} (\omega) - N (\omega) \big| \, \mathrm{d} \mathbf{P} (\omega) \to 0 \mbox{ as } t \to + \infty.</math>
 
===Corollary: convergence theorem for continuous martingales===
Let ''M''&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;&times;&nbsp;Ω&nbsp;→&nbsp;'''R''' be a [[sample continuous process|continuous]] martingale such that
 
:<math>\sup_{t > 0} \mathbf{E} \big[ \big| M_{t} \big|^{p} \big] < + \infty</math>
 
for some ''p''&nbsp;&gt;&nbsp;1. Then there exists a random variable ''M''&nbsp;∈&nbsp;''L''<sup>p</sup>(Ω,&nbsp;'''P''';&nbsp;'''R''') such that ''M''<sub>''t''</sub>&nbsp;→&nbsp;''M'' as ''t''&nbsp;→&nbsp;+∞ both '''P'''-almost surely and in ''L''<sup>p</sup>(Ω,&nbsp;'''P''';&nbsp;'''R''').
 
==Discrete-time results==
Similar results can be obtained for discrete-time supermartingales and submartingales, the obvious difference being that no continuity assumptions are required. For example, the result above becomes
 
Let ''M''&nbsp;:&nbsp;'''N'''&nbsp;&times;&nbsp;Ω&nbsp;→&nbsp;'''R''' be a discrete-time martingale such that
 
:<math>\sup_{k \in \mathbf{N}} \mathbf{E} \big[ \big| M_{k} \big|^{p} \big] < + \infty</math>
 
for some ''p''&nbsp;&gt;&nbsp;1. Then there exists a random variable ''M''&nbsp;∈&nbsp;''L''<sup>p</sup>(Ω,&nbsp;'''P''';&nbsp;'''R''') such that ''M''<sub>''k''</sub>&nbsp;→&nbsp;''M'' as ''k''&nbsp;→&nbsp;+∞ both '''P'''-almost surely and in ''L''<sup>p</sup>(Ω,&nbsp;'''P''';&nbsp;'''R''')
 
=={{anchor|Convergence of conditional expectations: Lévy's zero-one law}}Convergence of conditional expectations: Lévy's zero–one law==
Doob's martingale convergence theorems imply that [[conditional expectation]]s also have a convergence property.
 
Let (Ω,&nbsp;''F'',&nbsp;'''P''') be a [[probability space]] and let ''X'' be a random variable in ''L''<sup>1</sup>. Let ''F''<sub>∗</sub>&nbsp;=&nbsp;(''F''<sub>''k''</sub>)<sub>''k''∈'''N'''</sub> be any [[filtration (abstract algebra)|filtration]] of ''F'', and define ''F''<sub>∞</sub> to be the minimal [[sigma algebra|''&sigma;''-algebra]] generated by (''F''<sub>''k''</sub>)<sub>''k''∈'''N'''</sub>. Then
 
:<math>\mathbf{E} \big[ X \big| F_{k} \big] \to \mathbf{E} \big[ X \big| F_{\infty} \big] \mbox{ as } k \to \infty</math>
 
both '''P'''-almost surely and in ''L''<sup>1</sup>.
 
This result is usually called '''Lévy's zero–one law'''. The reason for the name is that if ''A'' is an event in ''F''<sub>∞</sub>, then the theorem says that <math>\mathbf{P}[ A | F_{k} ] \to \mathbf{1}_A </math> almost surely, i.e., the limit of the probabilities is 0 or 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be. This sounds almost like a [[tautology (logic)|tautology]], but the result is still non-trivial. For instance, it easily implies [[Kolmogorov's zero–one law]], since it says that for any [[tail event]] ''A'', we must have <math>\mathbf{P}[ A ] = \mathbf{1}_A </math> almost surely, hence <math>\mathbf{P}[ A ]\in\{0,1\}</math>.
 
==See also==
*[[Backwards martingale convergence theorem]]
 
{{No footnotes|date=January 2012}}
==References==
* {{cite book
| last = Øksendal
| first = Bernt K.
| authorlink = Bernt Øksendal
| title = Stochastic Differential Equations: An Introduction with Applications
| edition = Sixth
| publisher=Springer
| location = Berlin
| year = 2003
| id = ISBN 3-540-04758-1
}} (See Appendix C)
 
* {{cite book
| last = Durrett
| first = Rick
| title = Probability: theory and examples
| edition = Second
| publisher=Duxbury Press
| year = 1996
| id = ISBN 978-0-534-24318-0
}}
 
[[Category:Probability theorems]]
[[Category:Martingale theory]]

Latest revision as of 08:51, 25 May 2014


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