Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a probability space and $\sigma:\Omega\to\Omega$ be an ergodic, invertible and measure preserving transformation. Consider a family of column stochastic matrices $(A_\omega)_{\omega\in\Omega}$ with a one-dimensional top Oseledets space. I am interested in how quickly these matrices converge to their limit under large compositions of $A$. In particular, let's suppose that $A_\omega\cdots A_{\sigma^n\omega}\to B_\omega$ as $n\to\infty$. Here $B_\omega$ would be a column stochastic matrix with the same columns. How would one estimate $$\|A_\omega\cdots A_{\sigma^n\omega}-B_\omega\|.$$ In the situation that $\Omega=\{\omega\}$ this difference is related to the second eigenvalue of $A_\omega=A_{\omega_0}$, however, in the random setting I can't seem to find a reference addressing this question. I would guess that this convergence would be exponential and somehow dependent on the second Lyapunov exponents. Any help would be appreciated.
1 Answer
Let us first observe what will happen in the simplest case when the random variables $A_\omega,\dots,A_{\sigma^n\omega}$ are independent and uniformly distributed on a finite set of matrices (the Bernoulli process where $A_\omega$ is a function of the first coordinate). In this case, we will be able to calculate the asymptotic of even moments of $\|A_\omega\dots A_{\sigma^n\omega}-B_\omega\|$. For generality and simplicity, instead of assuming that the matrices $A_\omega$ are stochastic matrices, we shall assume that $A_\omega$ is an arbitrary matrix and we shall obtain asymptotic even moments for the norm $\|A_\omega\dots A_{\sigma^n\omega}\|$.
Suppose that $A_1,\dots,A_r\in M_n(K)$ are real,complex,or quaternionic matrices. For $1\leq p\leq\infty$, define the $L_p$-spectral radius of $A_1,\dots,A_r$ to be $\rho_p(A_1,\dots,A_r)=\overline{\text{lim}}_{N\rightarrow\infty}\|(\|A_{i_1}\dots A_{i_N}\|)_{i_1,\dots,i_N\in\{1,\dots n\}}\|_p^{1/n}.$
Then let $\Phi(A_1,\dots,A_r):M_n(K)\rightarrow M_n(K)$ be the completely positive operator defined by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+A_rXA_r^*$.
Lemma: $\rho(A_1\otimes B_1+\dots+A_r\otimes B_r)\leq\rho_p(A_1,\dots,A_r)\rho_q(B_1,\dots,B_r)$ whenever $p,q\in[1,\infty]$ and $\frac{1}{p}+\frac{1}{q}=1$.
The above lemma is a straightforward application of Holder's inequality.
Theorem: $\rho_2(A_1,\dots,A_r)=\rho(\Phi(A_1,\dots,A_r))^{1/2}$.
Proof: In this proof, recall that the Schatten $p$-norm of a matrix $X$ is defined as $\|X\|_p=\|(\sigma_1(X),\dots,\sigma_n(X))\|_p$ where $\sigma_1(X),\dots,\sigma_n(X)$ are the singular values of $X$. In particular, $\|X\|_2^2=\text{Tr}(XX^*)$ and if $X$ is positive semidefinite, then $\|X\|_1=\text{Tr}(X)$.
By using the above lemma, we know that $\rho(\Phi(A_1,\dots,A_r))^{1/2}\leq\rho_2(A_1,\dots,A_r)$.
For the converse direction, we observe that $\|\Phi(A_1,\dots,A_r)^N(I)\|_1=\text{Tr}(\Phi(A_1,\dots,A_r)^N(I))=\sum_{i_1,\dot,i_N}\text{Tr}(A_1\dots A_{i_N}A_{i_N}^*\dots A_{i_1}^*)=\sum_{i_1,\dots,i_N}\|A_1\dots A_{i_N}\|_2^2$.
Therefore, $\rho(\Phi(A_1,\dots,A_r))\geq\limsup_{N\rightarrow\infty}\|\Phi(A_1,\dots,A_r)^N(I)\|_1^{1/N}=\limsup_{N\rightarrow\infty}\sum_{i_1,\dots,i_N}\|A_1\dots A_{i_N}\|_2^2=\rho_2(A_1,\dots,A_r)^2.$
Q.E.D.
Define the tensor power of a matrix $A$ by setting $A^{\otimes 1}=A,A^{\otimes(n+1)}=A\otimes A^{\otimes n}=A^{\otimes n}\otimes A.$
Corollary: Let $q$ be a positive integer. Then $\rho_{2q}(A_1,\dots,A_r)=\rho(\Phi(A_1^{\otimes q},\dots,A_r^{\otimes q}))^{1/q}.$
The value $\rho_{2q}(A_1,\dots,A_r)$ tells us how quickly the $2q$-th moment of the norm $\|A_{i_1}\dots A_{i_N}\|$ grows.
I conjecture that we have the following characterization of the $L_2$-spectral radius in terms of random matrices. Suppose that $A_1,\dots,A_r$ are complex matrices. Let $U_1,\dots,U_r$ be random complex matrices with mean $0$ and where if $\alpha$ is a random entry in some $U_j$, then $\text{Var}(\text{Re}(\alpha))=\text{Var}(\text{Re}(i\alpha))=\frac{1}{2n}$, and suppose that all of these entries are independent. Then the eigenvalues of $A_1\otimes U_1+\dots+A_r\otimes U_r$ will be approximately uniformly distributed on a disk centered at $0$ with radius $\rho_2(A_1,\dots,A_r)$, and $\rho(A_1\otimes U_1+\dots+A_r\otimes U_r)\approx \rho_2(A_1,\dots,A_r).$
Suppose that $f:\Omega\rightarrow M_n(K)$ is a bounded measurable function. We shall associate $L_\infty(\mathbb{P})\otimes K^n$ with the set of all $L_\infty$ functions $g:\Omega\rightarrow K^n$. Given an $L_p$ function $g:\Omega\rightarrow K^n$, define $\hat{f}(g):\Omega\rightarrow K^n$ by letting $\hat{f}(g)(x)=f(x)\cdot g(\sigma(x))$. Then observe that $\hat{f}^N(g)(x)=f(x)f(\sigma(x))\dots f(\sigma^{N-1}(x))g(\sigma^N(x))$ for all $N>0$.
In this case, $\hat{f}:L_\infty(\mathbb{P})\otimes K^n\rightarrow L_\infty(\mathbb{P})\otimes K^n$ is a bounded linear operator, and $\|\hat{f}^N\|=\text{ess sup}\|f(x)f(\sigma(x))\dots f(\sigma^{N-1}(x))\|$. Therefore, $\rho(\hat{f})=\lim_{N\rightarrow\infty}(\text{ess sup}\|f(x)f(\sigma(x))\dots f(\sigma^{N-1}(x))\|)^{1/N}$.
