Abstract. We give a necessary and sufficient condition for the strong $(C,\alpha )$ law of large numbers with real order $\alpha >0$ for weighted sums of independent random variables satisfying the property $\alpha $-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 $(C,\alpha )$ law of large numbers for the sequence can also be characterized by the ergodic Hilbert transform.
AMS Subject Classification
(1991): 60F15; 47A35
strong law of large numbers,
Bourgain's return time theorem,
ergodic Hilbert transform,
universal sequence of weights
received 1.2.2021, revised 15.6.2021, accepted 30.6.2021. (Registered under 21/2021.)