Small deviations of series of weighted i.i.d. non-negative random variables with a positive mass at the origin

We study the small deviations of ∑j≥1λjXj and supj≥1λjXj, where {Xj} are i.i.d. non-negative random variables and 1/λj tends to ∞ faster than any power of j. The most interesting conclusions hold if P(X1=0)>0.