The Annals of Probability

The Stability of Large Random Matrices and Their Products

Joel E. Cohen and Charles M. Newman

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Let $A(1), A(2), \cdots$ be a sequence of independent identically distributed (i.i.d.) random real $n \times n$ matrices and let $x(t) = A(t)x(t - 1), t = 1, 2, \cdots$. Define $\bar{\lambda}_n = \sup\{\lim_{t \uparrow \infty}\|x(t)\|^{1/t}: 0 \neq x(0) \in R^n\}$ where $\|\cdot\|$ denotes, e.g. the Euclidean norm, providing the limit exists almost surely (a.s.) and is nonrandom, and define $\underline\lambda_n$ analogously with sup replaced by inf. If all $n^2$ entries of each $A(t)$ are i.i.d. standard symmetric stable random variables of exponent $\alpha$, then $\underline\lambda_n = \overline\lambda_n = \lambda_n(\alpha)$. In the standard normal case $(\alpha = 2), \lambda_n(2) = (2 \exp\lbrack\psi(n/2)\rbrack)^{1/2}$, where $\psi$ is the digamma function, and $n^{-1/2}\lambda_n(2) \rightarrow 1$; for $0 < \alpha < 2, (n \log n)^{-1/\alpha}\lambda_n(\alpha)$ converges to $\{2 \Gamma(\alpha) \sin(\alpha\pi/2)/\lbrack\alpha\pi\rbrack\}^{1/2}$. Criteria for stability $(\overline\lambda_n < 1)$ and instability $(\underline\lambda_n > 1)$ are investigated for more general distributions of $A(t)$. We obtain, for example, the general bound, $\overline\lambda_n \leq \{r\lbrack E(A(1)^T A(1))\rbrack\}^{1/2}$, where $A^T$ is the transpose of $A$ and $r$ denotes the spectral radius. In the case of independent entries of mean zero and common variance $s^2/n$, this leads to $\lim \sup_n \overline\lambda_n \leq s$. If the entries of $A(t)$ are i.i.d. and distributed as $W/n^{1/2}$ where $W$ is independent of $n$, has mean zero, variance $s^2$ and satisfies $E(\exp\lbrack iuW\rbrack) = O(|u|^{-\delta})$ as $|u| \uparrow \infty$ for some $\delta > 0$, then $\lim \inf_n\bar\lambda_n \geq s$. 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.

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Ann. Probab. Volume 12, Number 2 (1984), 283-310.

First available in Project Euclid: 19 April 2007

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Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60H25: Random operators and equations [See also 47B80] 92A17

Products of random matrices stability of ecosystems normal random matrices symmetric stable random matrices strong law central limit theorem stability of random linear ordinary differential equations Liapunov exponent


Cohen, Joel E.; Newman, Charles M. The Stability of Large Random Matrices and Their Products. Ann. Probab. 12 (1984), no. 2, 283--310. doi:10.1214/aop/1176993291.

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