Let us consider random matrices \(\{A_{k}\}_{k=1}^{N}\) of size \(d\times d\) independently chosen from the same ensemble and \(P_{N}= A_{N} A_{N-1} \cdots A_{1}\). Assuming the second moments of the diagonal entries of \(A^{†}_{k} A_{k}\) are finite, a multiplicative ergodic theorem states that the limiting matrix \(V_{d}= \lim_{N\rightarrow{\infty}} (P^{†}_{N}P_{N})^{1/(2N)}\) is well defined and, in fact, is a positive definite matrix with eigenvalues \(e^{\mu_{1}} \geq e^{\mu_{2}}\geq \cdots \geq e^{\mu_{d}}\). The real numbers \(\{\mu_{k}\}_{k=1}^{d}\) are called the Lyapunov exponents.
For large \(N\) and the eigenvalues of \(P^{†}_{N}P_{N}\) parameterized as \(x_{j}= e^{2N y_{j}}\), \(j=1,2, \dots,d\), the expression of the eigenvalue distribution of \((P^{†}_{N}P_{N})^{1/(2N)},\) when each \(A_{k}\) belongs to the ensemble of standard complex Gaussian matrices, has been done by G. Akemann et al. [J. Phys. A, Math. Theor. 47, No. 39, Article ID 395202, 35 p. (2014; Zbl 1327.60021)]. When \(N\) tends to infinity, the Lyapunov exponents are given by \(\mu_{j}=\frac{1}{2} \Psi (d-j+1), j=1,2 \dots,d\), where \(\Psi(x)\) denotes the digamma function, see [P. J. Forrester, J. Stat. Phys. 151, No. 5, 796–808 (2013; Zbl 1272.82020)].
In the paper under review, the author shows that methods known for the computation of \(\lim_{N\rightarrow{\infty}}\langle\mu_{i}\rangle\) can be used to compute the large \(N\) form of the variances of the exponents in the case of standard Gaussian random matrices with real, complex or real quaternion entries as well as extended to the general variance case for \(\mu_{1}\). Such a formulation allows for the computation of the variances associated with the distribution of \(\{y_{j}\}_{j=1}^{d}\) for large but finite \(N\) in other cases. First, the author focuses the attention on random matrices \(A_{k}= \Sigma^{1/2} G_{k},\) where \(\Sigma\) is a fixed \(d\times d\) positive definite matrix and \(G_{k}\) is a standard complex Gaussian random matrix. Next, the variance corresponding to the largest Lyapunov exponent, when \(G_{k}\) is a standard real, or standard real quaternion, Gaussian random matrix, is studied. Finally, the Lyapunov exponents and associated variances are computed for products of sub-blocks of random unitary matrices with real, complex or real quaternion entries and chosen with Haar measure.