Finite-Step Bounds for Iterated Correlation Matrices

Abstract

We establish finite-step probabilistic upper bounds on the contraction ratios $\rho_k = \Delta_{k+1}/\Delta_k$ for iterated Pearson correlation dynamics. Let $(P_k)_{k\ge 0}$ be the sequence generated by the Pearson update. Define $\Delta_k := \|P_{k+1}-P_k\|_F$, $\rho_k := \Delta_{k+1}/\Delta_k$ for $\Delta_k > 0$, and $\delta_k := \Delta_k/n$. Although $\Delta_k \to 0$ along convergent trajectories, the ratios $\rho_k$ may exceed unity in finitely many steps. This behavior is invisible to local linearization. Our main contribution is a probabilistic bounding framework that captures these finite-step expansions. We initialize $P_0$ with i.i.d. $\mathcal{U}[-1,1]$ entries and let $\mathbb{P}$ be the induced measure. For $k \ge 2$, we construct state-dependent bounds $B_p : \mathbb{R}_+ \to \mathbb{R}_+$ satisfying $\mathbb{P}(\rho_k \le B_p(\delta_k)) \ge p$. The functions $B^{\mathrm{q}}_p(\delta)$ are empirical conditional $p$-quantiles of $\log \rho_k$ given $\delta_k$ under logarithmic binning. Larger families $B^{\mathrm{TC}}_{p,\tau}(\delta)$ and $B^{\mathrm{tol}}_{p,\tau}(\delta)$ are obtained via multiplicative adjustments, yielding pointwise larger bounds that preserve the $\delta$-dependence. Validation on held-out trajectories confirms the bounds hold with empirical coverage matching nominal levels for all $n \in [3,2000]$. The baseline $0.95$-quantile bound $B^{\mathrm{q}}_{0.95}(\delta)$ yields two concrete results: $\mathbb{P}(\rho \le 1 \mid \delta \le 0.03) \ge 0.95$ uniformly in $n$, and $\mathbb{P}(\rho \le 1.7) \ge 0.95$ for 21 of 22 dimensions. The exception $n = 69$ attains $2.35$, revealing a rare extreme upper tail discontinuity not captured by asymptotic analysis. These are the first finite-step probabilistic bounds for Pearson correlation dynamics. The framework is fully reproducible with provided code and data.