Divergence rates for the number of rare numbers

Suppose thatX ₁,X ₂, ... is a sequence of i.i.d. random variables taking value inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY _n /EY _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 ofEY _n ]. These results extend earlier work by the author.