Asymptotic Expansions for i.i.d. Sums Via Lower-order Convolutions

Abstract

In this article, we introduce new asymptotic expansions for probability functions of sums of independent and identically distributed random variables. Results are obtained by efficiently employing information provided by lower-order convolutions. In comparison with Edgeworth-type theorems, advantages include improved asymptotic results in the case of symmetric random variables and ease of computation of main error terms and asymptotic crossing points. The first-order estimate can perform quite well against the corresponding renormalized saddlepoint approximation and, pointwise, requires evaluation of only a single convolution integral. While the new expansions are fairly straightforward, the implications are fortuitous and may spur further related work.

Keywords:

• Edgeworth expansions
• Local approximations
• Symmetric distributions
• Hermite polynomials
• Lattice distributions
• Convolutions
• Saddlepoint approximations
• Cumulants
• Normal distribution.

Mathematics Subject Classification:

• 60F05
• 60E05

Notes

X₁ is said to be a lattice random variable if there exist constants c and h > 0 such that X takes on values of the form c ± ih, , with probability one. The constant h is called a span of the distribution.