Divergence rates for the number of rare numbers

Suppose thatX₁,X₂, ... is a sequence of i.i.d. random variables taking value inZ⁺. Consider the random sequenceA(X)(X₁,X₂,...). LetY_n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY_n/E[Y_n] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y_n]. These results extend earlier work by the author.