A limit theorem for expectations conditional on a sum

Let ξ₁, ξ₂, ξ₃,... be a sequence of independent random variables, such that μ_j{equals colon}E[ξ_j], 0<α≤Var[ξ_j] and E[|ξ_j-μ_j|^2+δ] for some δ, 0<δ≤1, and every j≥1. If U and ξ₀ are two random variables such that E[ξ₀²]<∞ and E[|U|ξ₀²]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ_j:j≥1}, then under appropriate regularity conditions {Mathematical expression} where S_n{equals colon}ξ₁+ξ₂+⋯+ξ_n,μ_j{equals colon}E[ξ_j], s_n²{equals colon}Var[S_n], and c_n=O(s_n).