Divergence rates for the number of rare numbers
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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.
Key WordsRare numbers almost sure convergence subadditive
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