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A more general notion of weight called admissible is introduced and then aninvestigation is carried out on the a.e. convergence of weighted strong laws oflarge numbers and their applications to weighted one-sided ergodic Hilberttransforms on vector-valued $L_p$-spaces. Furthermore, the obtained results areapplied to the a.e. convergence of random modulated weighted ergodic series,which is also new in the classical setting.

Weighted Laws of Large Numbers and Weighted One-Sided ergodic Hilbert Transforms on Vector Valued $L_p$-spaces