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On the strong law of large numbers for martingales with operator norm. (English) Zbl 0943.60021

The strong law of large numbers for martingales with operator norm for vector-valued martingales is considered. Let \((S_{n},{\mathcal F}_{n},n\geq 0)\) be a martingale in the Euclidean space \(R^{m}\), \(m\geq 1\), \(S_0=0\), \(X_{i}=S_{i}-S_{i-1}.\) Let \((A_{n}\), \(n\geq 1)\) be a sequence of nonrandom linear operators mapping \(R^{m}\) into \(R^{d}\), \(1\leq d<\infty.\) Let \({\mathcal N}\) be the set of all strictly monotone sequences of positive integers increasing to infinity. The authors present the following assertion: Suppose that \(\|A_{n}X_{i}\|@>P>>0\) as \(n\to\infty\) for \(i\geq 1\). Then there exists a finite set of sequences \({\mathcal N}_{f}\subset{\mathcal N}^{+}\) depending only on the sequence \((A_{n},n\geq 1)\) and such that if the condition \(\sum_{j=1}^{\infty}E\|A_{n_{j+1}} (S_{n_{j+1}}-S_{n_{j}})\|^2<\infty\) holds for all sequences \((n_{j},j\geq 1)\in{\mathcal N}_{f}\), then a.s. \(\|A_{n}S_{n}\|\to 0\) as \(n\to\infty.\)
This result is applied to study the asymptotic behavior of solutions of stochastic recurrent equations. Next, the authors consider the equation \(Y_{n}=AY_{n-1}+Z_{n}\), \(n\geq 1\), \(Y_0\in R^{m}\), where \(A\) is an \(m\times m\) matrix, \((Z_{n},n\geq 1)\) is a martingale difference sequence in \(R^{m}.\) Let \(r\) be the spectral radius of the matrix \(A\) and let \(p\) be the greatest order of zeros of modulus \(r\) of the minimal polynomial of \(A.\) The following assertion is proved: Suppose that the following conditions hold: \(a_{n}r^{n} n^{p-1}\to 0\) as \(n\to\infty\) and \[ \sum_{i=1}^{\infty}\sup_{n\geq i}[a_{n}r^{n-i}(n+1-i)^{p-1}]^2 E\|Z_{i}\|^2 <\infty, \] where \((a_{n},n\geq 1)\) is a bounded sequence of positive constants. Then a.s. \(a_{n}\|Y_{n}\|\to 0\) as \(n\to\infty.\)

MSC:

60F15 Strong limit theorems
60G42 Martingales with discrete parameter
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