On almost sure stability of the solutions of stochastic recurrent equations (Q915260)

Necessary and sufficient conditions for almost sure (a.s.) convergence of Gaussian Markov sequences in \(R^ m\) are obtained (theorem 1). As a simple application of this assertion we obtain the main stability result of this paper. Theorem 2: Let \((X_ n\), \(n\geq 1)\) be a sequence of centered jointly Gaussian random variables which satisfies the following system of stochastic difference equations: \[ X_ n=a_{n1}X_{n- 1}+a_{n2}X_{n-2}+...+a_{nm}X_{n-m}+b_ nY_ n,\quad n\geq 1,\quad X_ 0=X_{-1}=...=X_{1-m}=0, \] where \((Y_ n\), \(n\geq 1)\) is a standard Gaussian sequence and \((a_{nk}\), \(n\geq 1\); \(k=1,2,...,m)\) and \((b_ n\), \(n\geq 1)\) are some given arrays of real numbers. Then \(X_ n\to X\) a.s. iff 1) \(X_ n\to^{P}X\); 2) for all nondecreasing and unbounded sequences of natural numbers \((n_ k\), \(k\geq 1)\) and every \(\epsilon >0\), \[ \sum_{k\geq 1}\exp (-\epsilon /d^ 2_ k)<\infty,\text{ where } d^ 2_ k=E(X_{n_ k}-E(X_{n_ k}| X_{n_{k-1}},X_{n_{k-1}- 1},...,X_{n_{k-1}-m+1}))^ 2. \] As corollaries, entropy criteria are presented for a.s. convergence of Gaussian and subgaussian solutions of the above recurrent equations.