Overview
This research focuses on the asymptotic expansion of binomial probabilities, specifically examining cases where the observation is a fixed integer distance from the upper mode. The study operates under conditions where the fractional component of the mean remains significant. A complete asymptotic expansion is presented, detailing its structure to all orders.
Research Context
The investigation concerns binomial probabilities. The specific conditions under which these probabilities are analyzed include a fixed integer distance from the upper mode and a discernible fractional part of the mean. This approach aims to provide a comprehensive asymptotic description of these probabilities.
Approach
The methodology involved deriving the complete asymptotic expansion. This expansion is developed to all orders. Its validity is demonstrated uniformly across compact subintervals of the success probability, ranging from $(0,1)$, and for shifts from the mode that remain bounded.
Findings
- The complete asymptotic expansion for binomial probabilities at a fixed integer distance from the upper mode was obtained to all orders.
- This expansion is uniform for the success probability within compact subintervals of $(0,1)$ and for bounded shifts from the mode.
- A key structural finding is that a logarithmic tail, observed in the standard local expansion, corresponds precisely to the binomial size-bias factor.
- Upon removal of this logarithmic tail, the remaining expansion constitutes a pure Appell expansion.
- A second, symmetric normalization in the expansion is governed by an even Appell sequence. This normalization recovers the classical Stirling prefactor.
- The derived coefficients explain the exact binomial mode rule.
- The expansion recovers central-binomial expansions as a specific instance.
- The study yields closed-form expressions for averaged oscillating coefficients.
Why This Matters
The derived complete asymptotic expansion provides a detailed mathematical understanding of binomial probabilities under specific conditions, clarifying the underlying structural components. The findings offer a unified framework that explains existing rules and expansions, potentially simplifying calculations for certain binomial probability scenarios.