Recursive Standard Deviation Dynamics: Bridging Statistical Theory and Economic Applications through Iterative Feedback

This study explores the dynamic behavior of standard deviation within a recursive feedback system, where the computed metric is iteratively appended to its dataset. Through extensive computational simulations, we demonstrate that standard deviation exhibits non-linear decay, asymptotically approaching a near-zero value while retaining a residual diversity floor (C ≈ 1.2×10⁻⁷). Initialized with a dataset [1, 2, ..., 10], the process reveals a two-phased decay: rapid initial decline (k ≈ 0.012) followed by gradual convergence. An exponential decay model, y(x) = 2.71e⁻⁰·⁰¹²ˣ + 1.2×10⁻⁷ (R² = 0.98), accurately captures this behavior, offering empirical insights into feedback-driven dynamics. These findings are contextualized within economic frameworks, including volatility modeling, adaptive policymaking, and machine learning-driven econometrics. The rapid decay mirrors financial market stabilization post-shock, while the diversity floor suggests inherent systemic risk, challenging complete risk elimination assumptions. For policymaking, the decay constant informs stabilization efficacy, and the asymptote sets realistic inflation volatility targets. In machine learning, preserving residual diversity enhances algorithmic robustness. This bridges statistical theory and economic practice, providing actionable insights for financial risk management, policy design, and trading strategies. Unlike traditional linear models (e.g., ARIMA), our recursive approach captures non-linear feedback effects, addressing a critical research gap. Future work should integrate stochastic elements for broader applicability.