A sequence of random variates , ,
... with finite means such that the conditional expectation of given , ,
, ..., is equal to , i.e.,
(Feller 1971, p. 210). The term was first used to describe a type of wagering in which the bet is doubled or halved after a loss or win, respectively. The concept of martingales is due to Lévy, and it was developed extensively by Doob.
A one-dimensional random walk with steps equally likely in either direction () is an example of a martingale.
, 
,
... with finite means such that the conditional expectation of 
given 
, 
,
, ..., 
is equal to 
, i.e.,
) is an example of a martingale.
