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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
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| :<math> \mathbb{E} (X_t | \mathcal{F}_{t-1}) = 0, a.s. </math>,
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| 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.
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| 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.
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| == References ==
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| * James Douglas Hamilton (1994), ''Time Series Analysis'', Princeton University Press. ISBN 0-691-04289-6
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| * James Davidson (1994), ''Stochastic Limit Theory'', Oxford University Press. ISBN 0-19-877402-8
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| {{Stochastic processes}}
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| [[Category:Stochastic processes]]
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| [[Category:Martingale theory]]
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| {{probability-stub}}
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Revision as of 12:05, 18 February 2014
The name of the writer is Jayson. What I adore doing is soccer but I don't have the time lately. Since I was eighteen I've been operating as a bookkeeper but quickly my spouse and I will start our own business. I've always loved residing in Mississippi.
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