Almost sure central limit theorems for random functions

Let {Xn, −∞ < n < ∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn = Σk=1∞Xk, and Tn = Tn(X1, ..., Xn) be a random function such that Tn = ASn + Rn, where supn E|Rn| < ∞ and Rn = o(hrn) a.s., or Rn = O(n1/2−2γ) a.s., 0 < γ < 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.