9.1 Definition and First Examples
Definition 9.1.1. A sequence \(\{X_n : n\geq 0\}\) of random variables with \(E\left |X_n\right |<\infty \) for every \(n\) is a martingale with respect to a sequence \(\{Y_n\}\) if \[E\left (X_{n+1} \mid Y_0, Y_1, \dots , Y_n\right ) = X_n \qquad \text {for every } n .\] If the equality is replaced by \(\geq \) the sequence is a submartingale, and by \(\leq \) a supermartingale. When no \(\{Y_n\}\) is named, the conditioning is on \(X_0,\dots ,X_n\) themselves.
Note. Read the definition as a statement about a gambler’s fortune. \(X_n\) is the capital after \(n\) plays; the condition says that, given the entire history, the expected capital after the next play is exactly what is held now. A martingale is a fair game, a submartingale one favourable to the player, a supermartingale one favourable to the house. The prefixes are the opposite way round from what most people first guess, and it is worth fixing them by that reading.
Proof. By the tower property and the defining relation, \[E\left (X_{n+1}\right ) = E\left [E\left (X_{n+1}\mid Y_0,\dots ,Y_n\right )\right ] = E\left (X_n\right ),\] so the expectation is the same at every step, and induction from \(n=0\) gives the result. □
Example 9.1.3 (Sums of independent centred variables). Let \(Z_1,Z_2,\dots \) be independent with \(E(Z_i)=0\), and put \(X_n = \sum _{i=1}^{n}Z_i\) with \(X_0=0\). Then \[E\left (X_{n+1}\mid Z_1,\dots ,Z_n\right ) = E\left (X_n + Z_{n+1}\mid Z_1,\dots ,Z_n\right ) = X_n + E\left (Z_{n+1}\right ) = X_n ,\] using that \(X_n\) is determined by the conditioning variables and that \(Z_{n+1}\) is independent of them. So the centred random walk is a martingale.
Example 9.1.4 (Products of independent unit-mean variables). Let \(W_1,W_2,\dots \) be independent, positive, with \(E(W_i)=1\), and put \(X_n = \prod _{i=1}^{n}W_i\), \(X_0=1\). Then \[E\left (X_{n+1}\mid W_1,\dots ,W_n\right ) = X_n\,E\left (W_{n+1}\right ) = X_n ,\] so \(\{X_n\}\) is a martingale. This is the model of a fortune multiplied by independent random returns, and it is the reason the next remark matters.
Remark. The second example is worth pausing on. Take \(W_i\) equal to \(2\) or \(0\) with probability \(\tfrac 12\) each, so \(E(W_i)=1\) and \(\{X_n\}\) is a martingale with \(E(X_n)=1\) for every \(n\). But \(X_n=0\) as soon as a single zero appears, which happens with probability one eventually, so \[X_n \underset {a.s.}{\longrightarrow } 0 \qquad \text {while}\qquad E(X_n)=1 \ \text { for all } n .\] The expectation does not converge to the expectation of the limit. This is the sharpest illustration in the course of why the convergence modes of Chapter 6 had to be separated: almost sure convergence says nothing on its own about means, and the extra hypothesis needed — uniform integrability — is exactly what the martingale convergence theorem below supplies in its stronger form.
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