Lim sup behavior of sums of geometrically weighted i.i.d. random variables

Geometrically weighted i.i.d. random variables {Y_n} which are bounded above are shown to exhibit iterated logarithm type behavior. Specifically, if b > 1 and if the lower tail of the distribution of Y₁ approaches 0 fast enough, then lim sup_n→∞(b?1) Σⁿ_j=1b¹Y_j?bⁿ⁺¹=L, almost certainly, where L is the essential supremum of Y₁.