a product of mean-one variables as a martingale
Alongside the additive martingale (a running SUM of mean-zero shocks) lives its multiplicative twin: a running PRODUCT of independent mean-one factors. Picture wealth that gets multiplied each period by an independent random factor whose average is exactly 1 — a gross return that is fair in the multiplicative sense, neither growing nor shrinking your capital on average. That running product is a martingale, and this multiplicative form is the seed of likelihood ratios, exponential martingales, and the change-of-measure machinery that drives mathematical finance.
Let X_1, X_2, ... be independent non-negative random variables each with E[X_i] = 1, and set Y_n = X_1 * X_2 * ... * X_n with Y_0 = 1. Conditioning on the past F_n, the next factor is independent with mean 1, so E[Y_(n+1) given F_n] = Y_n * E[X_(n+1)] = Y_n * 1 = Y_n — the martingale equation, with multiplication replacing addition. (If instead each E[X_i] = m, then Y_n / m^n is the martingale.) A celebrated special case is the exponential martingale: take X_i = exp(theta * Z_i - K(theta)) where Z_i are i.i.d. and K is the cumulant generating function; each factor has mean one by the very definition of K, so the product is a martingale, and it is the workhorse behind Chernoff bounds and Wald's identity. The likelihood ratio of one model versus another, computed on growing data, is another product martingale under the true model.
Product martingales are also the cleanest illustration of the gap between almost-sure and mean convergence. A non-negative martingale always converges almost surely, so Y_n approaches some limit Y_infinity. But the limit's mean can collapse: if the factors are not uniformly integrable, you can have E[Y_n] = 1 for all n while Y_infinity = 0 almost surely — for example with X_i taking values 0 or 2 each with probability 1/2, a single zero factor annihilates the whole product forever. So 'multiplicatively fair every step' does NOT mean 'preserves your capital in the limit'. The mass quietly leaks away, a vivid warning that constant expectation is not the same as a non-vanishing limit.
Let each X_i be 0 or 2 with probability 1/2, so E[X_i] = 1. The product Y_n = X_1 * ... * X_n is a martingale with E[Y_n] = 1 for every n. Yet as soon as a single factor is 0 — which eventually happens almost surely — the product is pinned at 0 forever, so Y_n converges almost surely to 0. Constant mean 1, vanishing limit 0: the textbook gap between a.s. and mean convergence.
A product of mean-one factors is a martingale; if not uniformly integrable, its mean stays 1 while it collapses to 0.
Multiplicatively fair at every step does NOT preserve capital in the limit: a mean-one product can converge to 0 almost surely while its expectation stays 1. The factors must be non-negative for it to be a martingale.