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In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies the identity-
i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation. As is frequent in probability theory, the term was adopted from the language of gambling.
Somewhat more generally, a sequence Y1, Y2, Y3, ... is said to be a martingale with respect to another sequence X1, X2, X3, ... if
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for every n.
1 History
Originally, martingale referred to a class of betting strategies popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with both infinite wealth and infinite time at his disposal is guaranteed to eventually flip heads, the martingale betting strategy was seen as a sure thing by those who practiced it. Unfortunately, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck.
The concept of martingale in probability theory was introduced by Paul Pierre Lévy, and much of the original development of the theory was done by Doob . Part of the motivation for that work was to show the impossibility of successful betting strategies.
2 Examples of martingales
- Suppose Xn is a gambler's fortune after n tosses of a "fair" coin, where the gambler wins $1 if the coin comes up heads and loses $1 if the coin comes up tails. The gambler's conditional expected fortune after the next trial, given the history, is equal to his present fortune, so this sequence is a martingale.
- Let Yn = Xn2 − n where Xn is the gambler's fortune from the preceding example. Then the sequence { Yn : n = 1, 2, 3, ... } is a martingale. This can be used to show that the gambler's total gain or loss grows roughly as the square root of the number of steps.
- ( de Moivre's martingale) Now suppose an "unfair" or "biased" coin, with probability p of "heads" and probability q = 1 − p of "tails". Let
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- with "+" in case of "heads" and "-" in case of "tails". Let
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- Then { Yn : n = 1, 2, 3, ... } is a martingale with respect to { Xn : n = 1, 2, 3, ... }.
- Let Yn = P(A | X1, ... , Xn). Then { Yn : n = 1, 2, 3, ... } is a martingale with respect to { Xn : n = 1, 2, 3, ... }.
- ( Polya's urn) An urn initially contains r red and b blue marbles. One is chosen randomly. Then it is put back together with another one of the same colour. Let Xn be the number of red marbles in the urn after n iterations of this procedure, and let Yn=Xn/(n+r+b). Then the sequence { Yn : n = 1, 2, 3, ... } is a martingale.
- ( Likelihood-ratio testing in statistics) A population is thought to be distributed according either to a probability density f or another probability density g. A random sample is taken, the data being X1, ... , Xn. Let Yn be the "likelihood ratio"
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- (which, in applications, would be used as a test statistic). If the population is actually distributed according to the density f rather than according to g, then { Yn : n = 1, 2, 3, ... } is a martingale with respect to { Xn : n = 1, 2, 3, ... }.
- Suppose each amoeba either splits into two amoebas, with probability p, or eventually dies, with probability 1 − p. Let Xn be the number of amoebas surviving in the nth generation (in particular Xn = 0 if the population has become extinct by that time). Let r be the probability of eventual extinctionStochastic processes The Galton-Watson process is a stochastic process arising from Francis Galton's statistical investigation of the extinction of surnames. History There was concern amongst the Victorians that aristocratic surnames were becoming extinct. (Finding r as function of p is an instructive exercise. Hint: The probability that the descendants of an amoeba eventually die out is equal to the probability that either of its immediate offspring dies out, given that the original amoeba has split.) Then
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- is a martingale with respect to { Xn: n = 1, 2, 3, ... }.
- The number of individuals of any particular species in an ecosystem 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 biodiversityThe unified neutral theory of biodiversity (here UNTB) is a scientific hypothesis that aims to explain the relative abundance of species in ecological communities. The theory is named in analogy to the neutral theory of molecular evolution, to which it is.
