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 X_j,n, j ≤ n, where a_j,n = f(j/n) for some f ∈ C²[0, 1]. Under Cramer’s condition, we prove an integro-local limit theorem for P(S_n ∈ [x, x + Δn)) as x/n ∈ [m⁻, m⁺] for some m⁻, m⁺ and any sequence Δ_n tending to zero slowly enough. This result covers the whole scope of normal, moderate, and large deviations. For the stochastic process Y_n(t) corresponding to S₀, . . . , S_n we obtain a conditional functional limit theorem concerning convergence Y_n(t) to the Brownian bridge given the condition S_n ∈ [x, x + Δn).