An Abelian theorem with application to the conditional Gibbs principle

2013/02/06 by Zhansheng Cao, Cao, Zhansheng · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1302.1337

arXiv admin note: substantial text overlap with arXiv:1206.6951

arxiv created 2013/02/06 · arxiv updated 2013/02/07

Abstract

Let X₁,...,X_n be n independent unbounded real random variables which have common, roughly speaking, light-tailed type distribution. Denote by S₁ⁿ their sum and by π^a_n the tilted density of X₁, where a_n →∞ as n→ ∞. An Abelian type theorem is given, which is used to approximate the first three centered moments of the distribution π^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 (S₁ⁿ/n=a_n) with a_n→ ∞.

Citations

• A conditional limit theorem for random walks under extreme deviation

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