The Stability of Large Random Matrices and Their Products

Let A(1), A(2), ⋯ be a sequence of independent identically distributed (i.i.d.) random real n × n matrices and let x(t) = A(t)x(t - 1), t = 1, 2, ⋯. Define λ̄ₙ = ↑ ∞\|x(t)\|1/t: 0 ≠ x(0) ∈ Rⁿ\ where \|·\| denotes, e.g. the Euclidean norm, providing the limit exists almost surely (a.s.) and is nonrandom, and define λₙ analogously with sup replaced by inf. If all n² entries of each $A(t)$ are i.i.d. standard symmetric stable random variables of exponent α, then λₙ = λₙ = λₙ(α). In the standard normal case (α = 2), λₙ(2) = (2 expψ(n/2))1/2, where ψ is the digamma function, and n-1/2λₙ(2) → 1; for 0 < α < 2, (n log n)-1/αλₙ(α) converges to \2 Γ(α) sin(απ/2)/απ\1/2. Criteria for stability (λₙ < 1) and instability (λₙ > 1) are investigated for more general distributions of $A(t)$. We obtain, for example, the general bound, λₙ ≤ E(A(1)T A(1))\1/2, where AT is the transpose of A and r denotes the spectral radius. In the case of independent entries of mean zero and common variance s²/n, this leads to lim ₙ λₙ ≤ s. If the entries of $A(t)$ are i.i.d. and distributed as W/n1/2 where W is independent of n, has mean zero, variance s² and satisfies E(exp iuW) = O(|u|-δ) as |u| ↑ ∞ for some δ > 0, then lim infₙλₙ ≥ s. These conditions for the asymptotic stability or instability of a product of random matrices are of the form originally proposed by May for differential equations governed by a single random matrix. We give counterexamples to show that May's criteria for the system of linear ordinary differential equations that he considered are not valid in the generality originally proposed, nor are they valid for the related system of difference equations considered by Hastings. The validity of May's criteria for these systems under more restrictive hypotheses remains an open question.