Page 77 - Elements of Distribution Theory

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2.6 Martingales 63

additional games will still be c.A formal statement of this result for martingales is given in
the following theorem.

Theorem 2.10. Let {X 1 , X 2 ,...} be a martingale and let n and m be positive integers. If
n < m, then

E(X m |X 1 ,..., X n ) = X n
with probability 1.

Proof. Since {X 1 , X 2 ,...} is a martingale,
E[X n+1 |X 1 ,..., X n ] = X n with probability 1.
Note that
E[X n+2 |X 1 ,..., X n ] = E[E(X n+2 |X 1 ,..., X n , X n+1 )|X 1 ,..., X n ]

= E[X n+1 |X 1 ,..., X n ] = X n
with probability 1. Similarly,

E(X n+3 |X 1 ,..., X n ) = E[E(X n+3 |X 1 ,..., X n , X n+1 , X n+2 |X 1 ,..., X n ]
= E(X n+2 |X 1 ,..., X n ) = X n
with probability 1. Continuing this argument yields the result.

The martingale properties of a sequence X 1 , X 2 ,... can also be described in terms of
the differences
D n = X n − X n−1 , n = 1, 2,...
where X 0 = 0. Note that, for each n = 1, 2,..., (X 1 ,..., X n )isa one-to-one function of
(D 1 ,..., D n ) since
X m = D 1 + ··· + D m , m = 1, 2,..., n.

Suppose that {X 1 , X 2 ,...} is a martingale. Then, by Theorem 2.5,
E{D n+1 |D 1 ,..., D n }= E{E(D n+1 |X 1 ,..., X n )|D 1 ,..., D n }
= E{E(X n+1 − X n |X 1 ,..., X n )|D 1 ,..., D n }= 0, n = 1, 2,....
A sequence of real-valued random variables D 1 , D 2 ,... satisfying

E{D n+1 |D 1 ,..., D n }= 0, n = 1, 2,...
is said to be a sequence of martingale differences.
As noted above, if X 1 , X 2 ,... is a martingale, then X n can be interpreted as the fortune
of a gambler after a series of fair games. In the same manner, if D 1 , D 2 ,... is a martingale
difference sequence, then D n can be interpreted as the amount won by the gambler on the
nth game.

Example 2.25 (Gambling systems). Suppose that a gambler plays a series of fair games
with outcomes D 1 , D 2 ,... such that, if the gambler places a bet B n on the nth game, her
winnings are B n D n .For each n, the bet B n is a function of D 1 ,..., D n−1 , the outcomes of