 Solution

Azuma's inequality states that if $(M_i{)}_{i=0}^m$ is a martingale with $|M_i-M_{i-1}|\le 1$ almost surely, then for $t\ge 0$,For independent product coordinates $Z_1,\dots ,Z_m$, expose them successively and set $M_i=E[f(Z)\mid Z_1,\dots ,Z_i]$. If changing one coordinate changes $f$ by at most one, the possible conditional means at that exposure have diameter at most one, by averaging the same remaining independent coordinates. Therefore the martingale increments have absolute value at most one. It follows thatFor coordinate bounds $c_i$, the same argument gives denominator $2{\sum }_i{c}_i^2$. The argument extends to arbitrary independent measurable coordinates for an integrable function with these bounds, rather than requiring finite state sets.