# Martingale difference sequence

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{{Short description|Sequence in probability theory}}
In [probability theory](/source/probability_theory), a '''martingale difference sequence''' ('''MDS''') is related to the concept of the [martingale](/source/martingale_(probability_theory)).  A [stochastic series](/source/stochastic_process) ''X'' is an MDS if its [expectation](/source/expected_value) with respect to the past is zero. Formally, consider an adapted [sequence](/source/sequence) <math>\{X_t, \mathcal{F}_t\}_{-\infty}^{\infty}</math> on a [probability space](/source/probability_space) <math>(\Omega, \mathcal{F}, \mathbb{P})</math>. <math>X_t</math> is an MDS if it satisfies the following two conditions:

:<math> \mathbb{E} \left|X_t\right| < \infty </math>, and

:<math> \mathbb{E} \left[X_t | \mathcal{F}_{t-1}\right] = 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](/source/independence_(probability_theory)), yet most limit theorems that hold for an independent sequence will also hold for an MDS.

A special case of MDS, denoted as {''X''<sub>''t''</sub>,''<math> \mathcal{F}</math>''<sub>''t''</sub>}<sub>''0''</sub><sup><math>{\infty} </math></sup> is known as innovative sequence of ''S''<sub>''n''</sub>; where ''S''<sub>''n''</sub> and <math> \mathcal{F}_{t}</math> are corresponding to [random walk](/source/random_walk) and [filtration](/source/filtration_(probability_theory)) of the random processes <math>\{X_{t}\}_0^\infty </math>. 

In [probability theory](/source/probability_theory) innovation series is used to emphasize the generality of [Doob representation](/source/Doob_decomposition_theorem). In [signal processing](/source/signal_processing) the innovation series is used to introduce [Kalman filter](/source/Kalman_filter). The main differences of [innovation](/source/Innovation_(signal_processing))
terminologies are in the applications. The later application aims to introduce the nuance of samples to the model by random sampling.

== References ==
{{refbegin}}
* 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}} 
{{refend}}

{{Stochastic processes}}

Category:Martingale theory
Category:Signal processing

{{probability-stub}}

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