Recursive Standard Deviation Dynamics: Unveiling Asymptotic Stability And The 'Diversity Floor' In Feedback Systems

Abstract

Eyas Gaffar A. Osman*

This study introduces a novel computational experiment to explore the dynamic behavior of standard deviation within a recursive feedback system, where the computed metric is iteratively appended to its own dataset. Through extensive simulations (50,000 iterations), we demonstrate that standard deviation exhibits a distinct non-linear decay pattern: an initial rapid decline (decay constant k ≈ 0.012) followed by a gradual convergence. Crucially, the standard deviation does not vanish but asymptotically approaches a non-zero, minute value (C ≈ 1.2 × 10-7), which we term the "diversity floor." An exponential decay model, y(x) = 2.71e-0.12x + 1.2 × 10-7 (R2 = 0.98), accurately captures this two-phased behavior, offering a robust empirical framework for feedback-driven dynamics. Contextualized within economic frameworks, our findings provide critical insights. The rapid decay phase mirrors financial market stabilization post-shock, while the persistent "diversity floor" challenges the assumption of complete risk eradication, suggesting an inherent systemic uncertainty. For adaptive policymaking, the decay constant k quantifies stabilization efficacy, and the asymptote C informs realistic volatility targets. In machine learning, preserving residual diversity, as highlighted by C, is shown to prevent overfitting and enhance algorithmic robustness and generalization by maintaining a necessary degree of internal variability. This research bridges statistical theory and economic practice, offering actionable insights for financial risk management, policy design, and the development of more resilient economic algorithms. Unlike traditional linear models (e.g., ARIMA, VAR), our recursive approach uniquely captures endogenous, non-linear feedback effects, addressing a significant research gap. Future work will integrate stochastic elements for broader real-world applicability.

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