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We give a necessary and sufficient condition for the strong (C, α) law of large numbers with real order α > 0 for weighted sums of independent random variables satisfying the property α-WH analogous to, though weaker than, the Hartman's type property. In particular, if a sequence of random variables is two-sided, then the strong (C, α) law of large numbers for the sequence can also be characterized by the ergodic Hilbert transform. [ABSTRACT FROM AUTHOR]

We prove a multiparameter L logk L generalization of the Báez-Duarte Abelian ergodic theorem for positive linear contractions on L1, which allows the application of the local convergence principle of Sucheston's type. Next we establish a new weighted Abelian ratio ergodic theorem for Dunford-Schwartz operators on L1 with modulation by Besicovitch sequences. Moreover, this (one-parameter) result is generalized to the case of multiparameter operator averages, which allows the application of Fava's maximal ergodic inequality. [ABSTRACT FROM AUTHOR]

This paper projects another affine case study in the program of analyzing multiparameter a.e. convergence, based on the Sucheston's type convergence principles. An affine semigroup is considered as a natural extension of strongly continuous semigroups of linear operators on Lp spaces. We prove some affine extensions of multiparameter martingale theorems, multiparameter ergodic theorems, and multiparameter ergodic theorems for the so-called nonlinear sums. Moreover, an affine (nonlinear) generalization is given of Berkson- Bourgain-Gillespie's theorem concerning the connection between the ergodic Hilbert transform and the ergodic theorem for power-bounded invertible linear operators on Lp (1 < p < ∞) spaces. In addition, the random ergodic Hilbert transforms will be established. We improve the local ergodic theorem of McGrath concerning strongly continuous m-parameter semigroups of positive linear operators in a more general affine setting. We shall also show that the Sucheston convergence principle is also very effective even in yielding a multiparameter generalization of Starr's theorem. The final section includes some examples. [ABSTRACT FROM AUTHOR]