Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

6.8 MARTINGALES

In this section, we present an interesting property of a random sequence that will be useful in Chapter 7 where it is decomposed into two distinct components. Without loss of generality, assume the process starts at time t = 0.

Definition: Martingale Sequence Random sequence X[k] with is a martingale if

(6.51)

It is important to note that there is no expectation on the right-hand side: this particular conditional mean is the most recent random variable X[k−1] of the sequence. Rewriting this expression, observe that

(6.52)

where X[k−1] has been brought inside the expectation because it is part of the conditioning. This expression shows that on average the change in a martingale sequence from one time instant to the next is zero.

Example 6.11. Let Y[k] be a Bernoulli random sequence where a coin with p = P(H) is tossed repeatedly, and define X[k] be the net amount of a gambler's winnings at time k. When the coin shows H, the gambler wins a fixed amount c>0; otherwise the gambler loses the same amount c. Thus

(6.53)

Conditioning on all previous net winnings gives