A uniform limit theorem for predictive distributions

✍ Scribed by Patrizia Berti; Pietro Rigo

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 121 KB
• Volume: 56
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(01)00089-x

✦ Synopsis

Let {Fn} be a ÿltration, {Xn} an adapted sequence of real random variables, and { n} a predictable sequence of non-negative random variables with 1 ¿ 0. Set ÿn = n i=1 i and deÿne the random distribution functions Fn(t) = (1=ÿn) n i=1 i I {X i 6t} and Bn(t) = (1=ÿn) n i=1 i P(Xi 6 t|Fi-1). Under mild assumptions on { n}, it is shown that sup t |Fn(t) -Bn(t)| → 0, a.s. on the set {Fn or Bn converges uniformly}. Moreover, conditions are given under which Fn converges uniformly with probability 1.