An Abelian theorem with application to the conditional Gibbs principle

Original · EN

Let X₁,...,Xₙ be n independent unbounded real random variables which have common, roughly speaking, light-tailed type distribution. Denote by S₁ⁿ their sum and by πᵃⁿ the tilted density of X₁, where aₙ →∞ as n→ ∞. An Abelian type theorem is given, which is used to approximate the first three centered moments of the distribution πᵃⁿ. Further, we provide the Edgeworth expansion of n-convolution of the normalized tilted density under the setting of a triangular array of row-wise independent summands, which is then applied to obtain one local limit theorem conditioned on extreme deviation event (S₁ⁿ/n=aₙ) with aₙ→ ∞.