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