Absolutely Fair

An absolutely fair sequence is a sequence of integrable random variables , , ... with expectation value and

almost surely for every positive integer for which is defined (Feller 1971). Here the brackets denote expectation value and conditional expectation, respectively, and integrable means that for every . If represents the gain on the th trial, the condition says that the conditional expectation of the next gain is zero even after all earlier gains are known.

The partial sums and form a martingale, since

Conversely, if , , ... is a martingale, then and form an absolutely fair sequence, with (Feller 1971).

Independent random variables with zero expectation value provide examples. For instance, independent random variables taking and 1 with equal probability give a symmetric one-dimensional random walk through their partial sums. The gains need not be independent random variables. If and are independent random variables each taking and 1 with equal probability, then the two-term sequence , is absolutely fair but dependent. Observing forces , while gives or 2.

Zero expectation value for each individual gain is insufficient. If takes and 1 with equal probability and for all , then but . After the first trial the later gains are known, so this sequence is not absolutely fair.