# Existence and uniqueness of the asymptotic distribution of $x(k+1) = Ax(k) + v(k)$

Consider the linear discrete-time stochastic systems: $$\begin{equation} x_{k+1} = Ax_k + v_k, \end{equation}$$ with time-instants $$k \in \mathbb{N}$$, state $$x_k \in \mathbb{R}^n$$, stochastic process $$v_k \in \mathbb{R}^n$$, and matrix $$A \in \mathbb{R}^{n \times n}$$. The process $$v_k$$ is an i.i.d. multivariate stochastic processes with $$E[v_k]=\mathbf{0}$$ and finite covariance matrix $$R_{1} := E[v_k v_k^T] \in \mathbb{R}^{n \times n}$$, $$R_{1} > 0$$ (positive definite). The initial state $$x_1$$ is a random vector with $$E[x_1]=\mathbf{0}$$ and finite covariance matrix $$R_0 \in \mathbb{R}^{n \times n}$$, $$R_0 >0$$. The process $$v_k$$, $$k \in \mathbb{N}$$ and the initial condition $$x_1$$ are mutually independent.

What I know is that if the eigenvalues of $$A$$ are inside the open unit disk, $$x_1 \sim \mathcal{N}(\mathbf{0},R_0)$$, and $$v_k \sim \mathcal{N}(\mathbf{0},R_1)$$, the process $$x_k$$ converges in distribution to $$\mathcal{N}(\mathbf{0},P)$$, where $$P$$ denotes the unique solution of $$APA^T-P = R_1$$.

My question is if we can conclude something about the asymptotic distribution of $$x_k$$ if $$x_1$$ and $$v_k$$ are not Gaussian. I know that even if they are not Gaussian the asymptotic mean is zero and the asymptotic covariance matrix is given by the same $$P$$. However, does having zero mean and finite covariance matrix imply the existence and uniqueness of an asymptotic distribution?

I have tried simulations with different distribution for $$x_1$$ and $$v_k$$, and as long as $$E[x_1]=\mathbf{0}$$ and $$E[v_k]=\mathbf{0}$$ with finite covariance matrices, I get unique asymptotic distributions.

Any hint or reference about this? Cheers, Carlos.