A uniform limit theorem for predictive distributions
Let \(\{F_n\}\) be a filtration, \(\{X_n\}\) an adapted sequence of real random variables, and \(\{\alpha_n\}\) a predictable sequence of nonnegative random variables with \(\alpha_1>0.\) The main purpose of this paper is to prove that \[ \sup_t |G_n(t)-B_n(t)|\to 0,\;\text{a.s. on the set \{\(G_n\) or \(B_n\) converges uniformly\}}, \] under mild conditions on \(\{\alpha_n\}\) of the type \(\sum_0^{\infty} \alpha_i=\infty\), \(\sum_0^{\infty} \alpha_i^2\beta_i^{-2}<\infty\) a.s. for two random distribution functions \[ G_n(t)=\beta_n^{-1}\sum_1^n \alpha_i I_{\{X_i\leq t\}}, \quad G_n(t)=\beta_n^{-1}\sum_1^n \alpha_i P(X_i\leq t\mid F_{i-1}),\quad \beta_n=\sum_1^n \alpha_i. \]
scientific article; zbMATH DE number 1796668
scientific article