Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form
Abstract
Let X∼ Bin(N,p) and let c=Nr be a prescribed centre with r≠ p. The finite formula for E|X−c| is classical, and the existence of a complete large-deviation expansion in powers of N−1 follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in closed form for the stop-loss tail produced by the absolute deviation, with the lattice defect of Nr retained explicitly. The construction combines three standard ingredients: Bernoulli polynomials for the logarithm of successive binomial masses, the exponential Bell recursion, and Eulerian polynomials for weighted geometric sums. The k-th coefficient has denominator exactly (1−ρ)2k+2, where ρ is the exponential tilt; hence the effective expansion parameter is 1/(N(1−ρ)2), the lattice weighted analogue of the parameter in the incomplete gamma expansion. The case r=1/2 gives the fold at the origin of a biased ±1 random walk, and its parity split is identified with the lattice defect of the fold.
Related
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BibTeX
@misc{elezovic2026absolute,
title = {{Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form}},
author = {Neven Elezović},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19064},
eprint = {2609.19064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19064},
}
RIS
TY - UNPB
TI - Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form
AU - Neven Elezović
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19064
UR - https://arxiv.org/abs/2609.19064
AB - Let X∼ Bin(N,p) and let c=Nr be a prescribed centre with r≠ p. The finite formula for E|X−c| is classical, and the existence of a complete large-deviation expansion in powers of N^−1 follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in closed form for the stop-loss tail produced by the absolute deviation, with the lattice defect of Nr retained explicitly. The construction combines three standard ingredients: Bernoulli polynomials for the logarithm of successive binomial masses, the exponential Bell recursion, and Eulerian polynomials for weighted geometric sums. The k-th coefficient has denominator exactly (1−ρ)^2k+2, where ρ is the exponential tilt; hence the effective expansion parameter is 1/(N(1−ρ)^2), the lattice weighted analogue of the parameter in the incomplete gamma expansion. The case r=1/2 gives the fold at the origin of a biased ±1 random walk, and its parity split is identified with the lattice defect of the fold.
ER -