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In [[probability theory]], a '''martingale difference sequence (MDS)''' is related to the concept of the [[martingale (probability theory)|martingale]]. A [[stochastic process|stochastic series]] ''X'' is an MDS if its [[expected value|expectation]] with respect to the past is zero. Formally, consider an adapted sequence <math>\{X_t, \mathcal{F}_t\}_{-\infty}^{\infty}</math> on a probability space <math>(\Omega, \mathcal{F}, \mathbb{P})</math>. <math>X_t</math> is an MDS if it satisfies the following two conditions:
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:<math> \mathbb{E} |X_t| < \infty </math>, and
 
:<math> \mathbb{E} (X_t | \mathcal{F}_{t-1}) = 0, a.s. </math>,
 
for all <math>t</math>. By construction, this implies that if <math>Y_t</math> is a martingale, then <math>X_t=Y_t-Y_{t-1}</math> will be an MDS—hence the name.
 
The MDS is an extremely useful construct in modern probability theory because it implies much milder restrictions on the memory of the sequence than [[independence (probability theory)|independence]], yet most limit theorems that hold for an independent sequence will also hold for an MDS.
 
 
== References ==
* James Douglas Hamilton (1994),  ''Time Series Analysis'', Princeton University Press. ISBN 0-691-04289-6
* James Davidson (1994), ''Stochastic Limit Theory'', Oxford University Press. ISBN 0-19-877402-8
 
{{Stochastic processes}}
[[Category:Stochastic processes]]
[[Category:Martingale theory]]
{{probability-stub}}

Latest revision as of 09:34, 6 July 2014

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