Martingale difference sequence
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probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, a martingale difference sequence (MDS) is related to the concept of the martingale. A stochastic series ''X'' is an MDS if its expectation with respect to the past is zero. Formally, consider an adapted sequence \_^ on a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
(\Omega, \mathcal, \mathbb). X_t is an MDS if it satisfies the following two conditions: : \mathbb \left, X_t\ < \infty , and : \mathbb \left \mathcal_\right= 0, a.s. , for all t. By construction, this implies that if Y_t is a martingale, then X_t=Y_t-Y_ 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 Independence is a condition of a nation, country, or state, in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of ...
, yet most limit theorems that hold for an independent sequence will also hold for an MDS. A special case of MDS, denoted as ''0'' is known as innovative sequence of ''S''''n''; where ''S''''n'' and \mathcal_ are corresponding to
random walk In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a rand ...
and
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
of the random processes \_0^\infty . In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
innovation series is used to emphasize the generality of Doob representation. In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
the innovation series is used to introduce
Kalman filter In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unk ...
. The main differences of
innovation Innovation is the practical implementation of ideas that result in the introduction of new goods or service (economics), services or improvement in offering goods or services. ISO TC 279 in the standard ISO 56000:2020 defines innovation as "a n ...
terminologies are in the applications. The later application aims to introduce the nuance of samples to the model by random sampling.


References

* James Douglas Hamilton (1994), ''Time Series Analysis'', Princeton University Press. * James Davidson (1994), ''Stochastic Limit Theory'', Oxford University Press. Martingale theory Signal processing {{probability-stub