1.10. Suppose there exist an

a, 0 < a < 2, a =1= 1, and two constants A and A2 such that 1

'rIn, 'rIx > 0: Al A2 -::::: 0O{IXnl > x} < -. xa -xa If a > 1 , suppose also that f (Xn ) o for each n Then for any sequence {an}

increasing to infinity, we have

[ThI~ ~esult, due to P. Levy and MarcmkiewIcz, was stated WIth a superfluous CO~dItIOn on {all}' Proceed as in Theorem 5.3.3 but truncate XrJ at arJ ; direct estImates are easy.]