Moderate deviations and Erdös–Rényi–Shepp laws for weighted sums

Let X0,X1, . . . be i.i.d. random variables with E(X0) = 0, E(X2 0) = 1 and E(exp{tX0}) < ∞ for any |t | < t0. We prove that the weighted sums V (n) = ∞j =0 aj (n)Xj , n 1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös–Rényi–Shepp laws for the weighted sums.  2003 Elsevier Inc. All rights reserved