Editing Martingale (probability theory) (section)

==Examples of martingales==
* An unbiased [[random walk]], in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where at each time step a move to the right or left is equally likely. 
* A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair. The gambler is playing a game of [[coin flipping]]. Suppose ''X<sub>n</sub>'' is the gambler's fortune after ''n'' tosses of a [[fair coin]], such that the gambler wins $1 if the coin toss outcome is heads and loses $1 if the coin toss outcome is tails.  The gambler's conditional expected fortune after the next game, given the history, is equal to his present fortune. This sequence is thus a martingale.
* Let ''Y<sub>n</sub>'' = ''X<sub>n</sub>''<sup>2</sup> − ''n'' where ''X<sub>n</sub>'' is the gambler's fortune from the prior example.  Then the sequence {''Y<sub>n</sub>'' : ''n'' = 1, 2, 3, ... } is a martingale.  This can be used to show that the gambler's total gain or loss varies roughly between plus or minus the [[square root]] of the number of games of coin flipping played.
* [[Abraham de Moivre|de Moivre]]'s martingale: Suppose the [[Fair coin|coin toss outcomes are unfair]], i.e., biased, with probability ''p'' of  coming up heads and probability ''q''&nbsp;=&nbsp;1&nbsp;−&nbsp;''p'' of tails.  Let

::<math>X_{n+1}=X_n\pm 1</math>
:with "+" in case of "heads" and "−" in case of "tails".  Let

::<math>Y_n=(q/p)^{X_n}</math>

:Then {''Y<sub>n</sub>'' : ''n'' = 1, 2, 3, ... } is a martingale with respect to {''X<sub>n</sub>'' : ''n'' = 1, 2, 3, ... }. To show this
::<math>
\begin{align}
E[Y_{n+1} \mid X_1,\dots,X_n] & = p (q/p)^{X_n+1} + q (q/p)^{X_n-1} \\[6pt]
& = p (q/p) (q/p)^{X_n} + q (p/q) (q/p)^{X_n} \\[6pt]
& = q (q/p)^{X_n} + p (q/p)^{X_n} = (q/p)^{X_n}=Y_n.
\end{align}
</math>
* [[Pólya's urn]] contains a number of different-coloured marbles; at each [[iterative method|iteration]] a marble is randomly selected from the urn and replaced with several more of that same colour. For any given colour, the fraction of marbles in the urn with that colour is a martingale. For example, if currently 95% of the marbles are red then, though the next iteration is more likely to add red marbles than another color, this bias is exactly balanced out by the fact that adding more red marbles alters the fraction much less significantly than adding the same number of non-red marbles would.
* [[Likelihood-ratio test]]ing in [[statistics]]: A random variable ''X'' is thought to be distributed according either to probability density ''f'' or to a different probability density ''g''.  A [[random sample]] ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> is taken.  Let ''Y''<sub>''n''</sub> be the "likelihood ratio"

::<math>Y_n=\prod_{i=1}^n\frac{g(X_i)}{f(X_i)}</math>

: If X is actually distributed according to the density ''f'' rather than according to ''g'', then {''Y<sub>n</sub>'' :''n''=1, 2, 3,...} is a martingale with respect to {''X<sub>n</sub>'' :''n''=1, 2, 3, ...}

[[Image:Martingale1.svg|thumb|250px|Software-created martingale series]]
* In an [[ecological community]], i.e. a group of species that are in a particular trophic level, competing for similar resources in a local area, the number of individuals of any particular species of fixed size is a function of (discrete) time, and may be viewed as a sequence of random variables. This sequence is a martingale under the [[unified neutral theory of biodiversity and biogeography]].
* If { ''N<sub>t</sub>'' : ''t''&nbsp;≥&nbsp;0 } is a [[Poisson process]] with intensity ''λ'', then the compensated Poisson process {&nbsp;''N<sub>t</sub>''&nbsp;−&nbsp;''λt'' : ''t''&nbsp;≥&nbsp;0&nbsp;} is a continuous-time martingale with [[Classification of discontinuities|right-continuous/left-limit]] sample paths.
* [[Wald's martingale]]
* A <math>d</math>-dimensional process <math>M=(M^{(1)},\dots,M^{(d)})</math> in some space <math>S^d</math> is a martingale in <math>S^d</math> if each component <math>T_i(M)=M^{(i)}</math> is a one-dimensional martingale in <math>S</math>.