Asymptotic Expansions for i.i.d. Sums Via Lower-order Convolutions |
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Authors: | Kenneth S. Berenhaut James W. Chernesky Jr. Ross P. Hilton |
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Affiliation: | 1. Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina, USAberenhks@wfu.edu;3. Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina, USA;4. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA |
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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. |
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Keywords: | Edgeworth expansions Local approximations Symmetric distributions Hermite polynomials Lattice distributions Convolutions Saddlepoint approximations Cumulants Normal distribution. |
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