An Abelian theorem with application to the conditional Gibbs principle

Abstract

Let $X_1,...,X_n$ be $n$ independent unbounded real random variables which have common, roughly speaking, light-tailed type distribution. Denote by $S_1^n$ their sum and by ${\pi }^{a_n}$ the tilted density of $X_1$, where $a_n\to \infty$ as $n\to \infty$. An Abelian type theorem is given, which is used to approximate the first three centered moments of the distribution ${\pi }^{a_n}$. 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 $(S_1^n/n=a_n)$ with $a_n\to \infty$.

arXiv admin note: substantial text overlap with arXiv:1206.6951

References in corpus (1)

• A conditional limit theorem for random walks under extreme deviation