Large Deviations of Weighted Sums of Independent Identically Distributed Random Variables with Functionally-Defined Weights

Abstract

Let \( {S}_n=\sum \limits_{j=1}^n{a}_{j,n}{X}_{j,n} \) be a weighted sum with independent, identically distributed steps Xj,n, j ≤ n, where aj,n = f(j/n) for some f ∈ C2[0, 1]. Under Cramer’s condition, we prove an integro-local limit theorem for P(Sn ∈ [x, x + Δn)) as x/n ∈ [m−, m+] for some m−, m+ and any sequence Δn tending to zero slowly enough. This result covers the whole scope of normal, moderate, and large deviations. For the stochastic process Yn(t) corresponding to S0, . . . , Sn we obtain a conditional functional limit theorem concerning convergence Yn(t) to the Brownian bridge given the condition Sn ∈ [x, x + Δn).