Suppose we have a stochastic linear process:
$$x_{k+1} = Ax_{k} + Bw_{k} \qquad \text{with} \qquad x_{0} = a\in\mathbb{R}^{n}$$
Here, $A,B$ are known matrices and $a$ is a known vector. Moreover, the eigenvalues of $A\in\mathbb{R}^{n\times n}$ are in the open unit disk. Finally, the elements of the sequence $\{w_{k}\}_{k\in\mathbb{N}}$ are iid random variables uniformly distributed on the $m$-dimensional unit cube.
- What would be the distribution of $x_{k}$ as $k\to\infty$?
It is a bell-shaped distribution, but for sure it is not Gaussian as it must have a compact support---due to the fact that $w_{k}$ are compactly supported and the eigenvalues of $A$ are in the open unit disk.
- As an example, lets take a scalar system with $A,B = 1/2$, and $x_0 = 0$. For $k=0$, $x_0$ has a dirac distribution. For $k=1$, we have a uniform distribution supported on $[-\frac{1}{2},\frac{1}{2}]$. For $k=2$ we would get some trapezoidal distribution supported on $[-\frac{3}{4},\frac{3}{4}]$. For each $k$, the distribution of $x_{k}$ is some piecewise polynomial function compactly supported which indeed looks increasingly bell-shaped. Even for this case, I can't really figure out what the limit distribution would be.
Thanks in advance.
Edit 1:
The linear recursion can indeed be arranged as:
$$ x_{N} = \sum^{N-1}_{k=0} A^{k}Bw_{N-k-1}$$
with $w_{k}$ being, for $k\in\{0,\ldots,N-1\}$, being independent and identically uniformly distributed on $[-1,1]^{m}$. Thus, for $x_{N}$ with $N\to\infty$ we are looking at some form of the central limit theorem, where $x_{\infty}$ can not possibly be normally distributed. As a plus I would like to know if the family of distributions generated by the sum for each $N$ has some sort of name (the first is a multidimensional trapezoidal distribution, the second is piecewise cuadratic, then piecewise cubic,...).
Thanks again in advance.