I am interested in solving the following matrix equation in variable $X$ for known matrices $A,B$: $$ X + AXA^\top + A^2X(A^\top)^2 + \cdots + A^nX(A^\top)^n = B.\tag{$*$} $$ Is there a standard approach to solving for $X$? If $A$ is a stable matrix, is equation ($*)$ related to a Lyapunov equation (detailed below) or an $n$-period approximation of the Lyapunov equation?
Lyapunov equation:
The discrete time Lyapunov equation in matrix variable $X$ can be written as $$ A X A^H - X + Q = 0 $$ where $Q$ is Hermitian and $A^H$ is the conjugate transpose of $A$. If $A$ is a stable matrix, then the solution is given by $$ X = \sum_{t=0}^\infty A^t Q (A^H)^t. $$
Not sure about convergence, but perhaps set up an iteration:
$$X_{k+1} = B - (AX_kA^\top + A^2X_k(A^\top)^2 + \cdots + A^nX_k(A^\top)^n)$$