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.

DOI: 10.14232/actasm-021-271-y

AMS Subject Classification (1991): 60F15; 47A35

Keyword(s): strong law of large numbers, weak homogeneity, Bourgain's return time theorem, sampling scheme, Doob scheme, ergodic Hilbert transform, universal sequence of weights

received 1.2.2021, revised 15.6.2021, accepted 30.6.2021. (Registered under 21/2021.)