Unparalleled instances of prolifickness, random walks, and square root boundaries
Stefan Gerhold, Friedrich Hubalek
TL;DR
The paper analyzes how stopping rules in a family-planning problem influence the total and average sex ratios, modeling births as a simple symmetric random walk $S_k$ and analyzing quantities like $R_n$ and $F_n$. It demonstrates that the total ratio converges to 1 almost surely under broad stopping rules, while family- or average-level metrics can deviate, with the magnitude governed by tail properties of stopping times. The authors derive limit laws for key strategies: '$p$ boys' yields Gaussian fluctuations and a half-limit for averages, while '$p$ boys more' produces heavy-tailed limits and non-Gaussian behavior; a square-root boundary strategy induces heavy-tailed tails and a distinct stable-limit regime. They also correct and extend classic results on one-sided square-root hitting times, and provide large-deviation principles via saddle-point analysis, connecting fluctuation theory, first-passage techniques, and actuarial-ruin-inspired methods to reveal how extreme stopping rules shape demographic-like statistics. Overall, the work highlights that even when a global ratio converges to unity, aggressive stopping rules can leave lasting, quantifiable imprints on both distributional limits and deviation probabilities, with rates controlled by tail exponents and boundary geometry.
Abstract
We revisit the problem of influencing the sex ratio of a population by subjecting reproduction of each family to some stopping rule. As an easy consequence of the strong law of large numbers, no such modification is possible in the sense that the ratio converges to 1 almost surely, for any stopping rule that is finite almost surely. We proceed to quantify the effects and provide limit distributions for the properly rescaled sex ratio. Besides the total ratio, which is predominantly considered in the pertinent literature, we also analyze the average sex ratio, which may converge to values different from 1. The first part of this note is largely expository, applying classical results and standard methods from the fluctuation theory of random walks. In the second part we apply tail asymptotics for the time at which a random walk hits a one-sided square root boundary, exhibit the differences to the corresponding two-sided problem, and use a limit law related to the empirical dispersion coefficient of a heavy-tailed distribution. Finally, we derive a large deviations result for a special stopping strategy, using saddle point asymptotics.