Probabilistic Analysis of the Random Spectral Radius for a Matrix Family

TL;DR

This work introduces the random spectral radius (RSR) as a probabilistic growth rate for products of matrices drawn from a finite family. By modeling length-$n$ products with IID sampling, the authors establish Law of Large Numbers and Central Limit Theorems for commuting and triangular matrix families, deriving explicit limiting values $\rho_\infty$ and Gaussian fluctuations with scale $n^{-1/2}$ (and Edgeworth refinements). They extend the analysis to small perturbations of diagonal matrices via perturbation theory and demonstrate robustness of the LLN/CLT under perturbations, while numerical experiments suggest similar typical-behavior generalizes to unconstrained matrix families. The results provide a probabilistic framework bridging deterministic extremal measures (JSR/LSR) and average-case dynamics, with implications for stability analysis of switching systems and refinement equations. The work also outlines future directions for nonnegative matrices and broader generalizations, linking to Lyapunov exponents and Perron–Frobenius theory.

Abstract

We investigate joint spectral characteristics of a family of matrices $\mathcal{F}$, associated with products in the semigroup generated by $\mathcal{F}$. In the literature, extremal measures such as the well-known joint spectral radius and the lower spectral radius have been extensively studied. However, these measures fail to capture the typical growth rate of matrix products, focusing instead on the worst and best-case scenarios. Nevertheless, when examining, for instance, a switching dynamical system, a probabilistic rate of growth, which characterizes typical trajectories, emerges as a highly intriguing and significant measure. In this article, we present, to the best of our knowledge, the first rigorous analysis of the random spectral radius. This joint spectral characteristic is computed on the set of length-$n$ products from a semigroup by random sampling according to a given probability measure. We establish asymptotic results-including a Law of Large Numbers and a Central Limit Theorem-for cases where the matrices are either diagonal (equivalently, commuting), upper- or lower-triangular, or small perturbations of diagonal matrices. Subsequently, we provide numerical evidence that the random spectral radius of arbitrary (that is structurally unconstrained) families of matrices exhibits asymptotic behavior similar to that of diagonal or nearly diagonal matrix families.

Figures (7)

• Figure 1: Histogram of $\rho_{800}(\mathcal{T}, \mathbb P_{\mathcal{T}})$ compared with the Gaussian pdf. Here, the family consists of two $3 \times 3$ real-valued matrices with $J = \{1\},$$\rho_\infty \approx 1.323194$ and $\sigma_\infty \approx 0.086861$.
• Figure 2: The histogram represents the empirical distribution, and its kernel density estimate (KDE) is shown as the green curve. These are compared against the theoretical limiting probability density function (pdf, orange curve) for $\xi^{(3)} = \rho_\infty \cdot \max_{j \in \{j_1, j_2, j_3\}} G_j$. (left) The components of $\boldsymbol{G}$ have equal variances and correlations. (right) The components of $\boldsymbol{G}$ have unequal variances and are uncorrelated.
• Figure 3: Univariate Edgeworth expansions: relative errors of $\mathbb{E}[\rho_n]$and $n\operatorname{Var}(\rho_n)$ for 1) (top) and 2) (bottom) models. All errors decay at the theoretically predicted rates or better.
• Figure 4: Validation of multivariate Edgeworth expansions via Monte Carlo simulation with $N_{\text{MC}} = 10^8$ samples per $n$-value. Note that, for $n>10^3$, entries of matrix products become more and more unreliable due to accumulation of errors.
• Figure 5: Histogram of $\rho_n(\mathcal{F},\mathbb P_{\mathcal{F}})$ for $n\in\{100,400\}$. The family consists of three $3 \times 3$ matrices with probabilities $\{0.3, 0.3, 0.4\}$ and $\mu = 0.3051690.5323040.671785^\top$ so $j_\star = 3$. In this case $\gamma \approx 10^{-3}$ has the same order of the discrepancy between $\rho_\infty = 1.95772960$ and the value obtained empirically (e.g., $\approx 1.95772380$ for $n=400$).

Theorems & Definitions (39)

• Remark 1
• Theorem 1: LLN
• proof
• Remark 2
• Theorem 2: CLT
• proof
• Theorem 3: LLN
• proof
• Theorem 4: CLT
• proof