The convergence of double-indexed weighted sums of martingale differences and its application. (English) Zbl 1474.60116
Summary: We investigate the complete moment convergence of double-indexed weighted sums of martingale differences. Then it is easy to obtain the Marcinkiewicz-Zygmund-type strong law of large numbers of double-indexed weighted sums of martingale differences. Moreover, the convergence of double-indexed weighted sums of martingale differences is presented in mean square. On the other hand, we give the application to study the convergence of the state observers of linear-time-invariant systems and present the convergence with probability one and in mean square.
MSC:
| 60G42 | Martingales with discrete parameter |
| 60F15 | Strong limit theorems |
| 60F25 | \(L^p\)-limit theorems |
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