A Law of Iterated Logarithm for Weighted Sums of Mixing Sequences of Random Variables

Chow and Teicher proved in (1) the following law of iterated logarithm for a type of weighted sums of random variables : If {X_j, j≧1} is i.i.d. random variables with EX_j=0, EX^2_j=1 and {a_j, j≦1} are real constants satisfying [numerisal formula] and [numerisal formula] for some C>O, then [numerisal formula] where [numerisal formula]. Recently, Tomkins proved in (8) a theorem of the same type for martingales. In this paper, we shall extend this theorem to that for φ-mixing sequence of random variables, under the conditions for {a_j, j≧1} which are stronger than those in (1). But, a part of examples in (1) is contained in our results