Martingale: In the gambling context the term at first referred to a system for recouping losses by doubling the stake after each loss has occured. The modern mathematical concept can also be seen in such terms as the mathematical formalization of the concept of a 'fair' game defined as one in which the expected size of the gambler's bank after nth game given the results of all earlier games is unchanged from the size after the (n- 1)th game, that is the gambler's average gain or loss after each game, given the earlier history of the games is zero. More formally the term refers to the stochastic process, or a series of random variables S1; S2; . . ., which is such the expected value of the ðn þ 1Þth random variable conditional on values of the first n, equals to the value of the nth random variable which can be given as

 

so that E(Sn) does not depend on the value of n.