Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ starting from an initial non-zero position $X_0$, where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb C^{d}$ are both constructed from i.i.d. zero mean circularly symmetric complex Gaussian variables whose mean and variance do not depend on $n$. Is it possible to show that $$\lim_{n\to\infty}\frac{1}{n}\log||X_n|| \geq 0?$$
Here's what I have.
Firstly, we see that for $\alpha_n,\beta_n\in\mathbb C^d$, \begin{equation}\lim_{n\to\infty}\frac{1}{n}\log||\alpha_n+\beta_n|| \leq \max\left\{ \lim_{n\to\infty}\frac{1}{n}\log||\alpha_n || \;, \;\lim_{n\to\infty}\frac{1}{n}\log||\beta_n|| \right\}\;(1)\end{equation} since $||\alpha_n +\beta_n||\leq 2\max\{ ||\alpha_n||, ||\beta_n ||\} $. Furthermore, $(1)$ holds with equality when \begin{equation} \min\left\{ \lim_{n\to\infty}\frac{1}{n}\log||\alpha_n || \;, \;\lim_{n\to\infty}\frac{1}{n}\log||\beta_n|| \right\} < \max\left\{ \lim_{n\to\infty}\frac{1}{n}\log||\alpha_n || \;, \;\lim_{n\to\infty}\frac{1}{n}\log||\beta_n|| \right\}. \;(2)\end{equation}
Considering the system under question, we have \begin{eqnarray}\lim_{n\to\infty}\frac{1}{n}\log||X_n|| &= &\lim_{n\to\infty}\frac{1}{n}\log||\mathbf A_n X_{n-1} + R_n|| \\&\leq& \max \left\{\lim_{n\to\infty}\frac{1}{n}\log||\mathbf A_n X_{n-1} + 0.5R_n|| ,\lim_{n\to\infty}\frac{1}{n}\log||0.5R_n|| \right\} \;(3) \\&=&\max \left\{\lim_{n\to\infty}\frac{1}{n}\log||X_n|| , 0 \right\} . \end{eqnarray}
If $$\lim_{n\to\infty}\frac{1}{n}\log||X_n|| \geq 0,$$ our result is reached trivially; if $$\lim_{n\to\infty}\frac{1}{n}\log||X_n|| = \lambda< 0,$$ we reach a contradiction because (from $(2)$) $(3)$ holds with equality, which gives $\lambda=0$.