Asymptotic behavior of solutions of stochastic recurrence equations in \(\mathbb{R}{}^ d\) (Q1177741)

Let the random sequence \((X_ n)\) in \(R^ d\) be the solution of the equation \[ X_{n+1}=AX_ n+B_{n+1}\Gamma_{n+1}, \qquad n\geq 0, \quad X_ 0=0, \] where \(A\), \((B_ n)\) are nonrandom matrices of dimension \(d\times d\); \((\Gamma_ n)\) is a sequence of independent Gaussian \(N(0,1)\)-distribution random vectors in \(R^ d\) (\(I\) is the identity matrix). We establish necessary and sufficient conditions for \[ \lim_{n\to\infty}C_ n X_ n=0 \hbox{ a.s.}\tag{1} \] and \(\overline{\lim}_{n\to\infty}\| C_ nX_ n\|<\infty\) a.s., where \((C_ n)\) is any fixed sequence of nondegenerate matrices, \(\|\cdot\|\) is the Euclidean norm of vectors in \(R^ d\). Let \((\lambda-\lambda_ 1)^{p_ 1},\dots,(\lambda-\lambda_ \nu)^{p_ \nu}\) be elementary divisors of the matrix \(\lambda I-A\), where \(\lambda_ 1,\dots,\lambda_ \nu\) are eigenvalues of the matrix \(A\) some of them may be equal; \(\sum_{i=1}^ \nu p_ i=d\). Denote \(r=\max_{1\leq i\leq\nu} |\lambda_ i|\), \(p=\max_{i\in G}p_ i\), and let \(G\subset\{1,2,\dots,\nu\}\) be such that \(i\in G\) if and only if \(|\lambda_ i|=r\). Then for (1) to hold it is sufficient that \[ \lim_{n\to\infty}\| C_ n\|^ 2_ E r^{2n}n^{2(p-1)} f(n)\ln\left( e+\sum_{k=1}^{n- 1}\min\{1,\ln(f(k+1)/f(k))\}\right) =0,\tag{2} \] where \(f(n)=\sum_{i=1}^ n\| B_ i\|^ 2_ E r^{-2i}(1-(i- 1)/n)^{2(p-1)}\) and \(\|\cdot\|_ E\) is the Euclidean norm of matrices in \(R^ d\). It is shown, that under some additional assumptions, condition (2) is necessary for the fulfilment of (1).