A uniform limit theorem for predictive distributions

Let {F-n} be a filtration, {X-n} an adapted sequence of real random variables, and {a(n)} a predictable sequence of non-negative random variables with alpha(1) > 0. Set beta(n) = Sigma(i=1)(n) alpha(i) and define the random distribution functions F-n(t) = (1/beta(n)) Sigma(i=1)(n) alpha(i)I{X-iless than or equal tot} and B-n(t) = (1/beta(n)) Sigma(i=1)(n) alpha(i)P(X-i less than or equal to t\Ft-1). Under mild assumptions on {alpha(n)}, it is shown that sup\F-n,(t) - B-n(t)\ --> 0, a.s. on the set {F-n or B-n converges uniformly}. Moreover, conditions are given under which F-n converges uniformly with probability 1.