Characterize the covariance-decay exponent as a function of the excitation exponent

Characterize rigorously the decay-rate exponent \(\gamma\) in the asymptotic covariance behavior \(p_k=O(k^{-\gamma})\) as a function of the self-regulating excitation-law exponent \(\beta\).

Background

The proposed excitation law sets the probing amplitude according to $\sigma_k=\sigma_0(p_k/p_0)^\beta$, where $p_k=\lambda_{\max}(P_k)$ is the largest eigenvalue of the recursive least-squares covariance matrix and $0<\beta<1$. Theorem 1 establishes that both $\sigma_k$ and $p_k$ converge to zero, while the parameter estimate converges to the true parameter and $\sigma_k/p_k\to\infty$.

Numerical simulations suggest that the covariance measure follows a power-law decay $p_k\approx k^{-\gamma}$, with $\gamma$ decreasing as $\beta$ increases. The unresolved problem is to prove and rigorously characterize this dependence, thereby quantifying the tradeoff between retained excitation and covariance-convergence rate.