Abstract
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 /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.
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Key, E.S. Divergence rates for the number of rare numbers. J Theor Probab 9, 413–428 (1996). https://doi.org/10.1007/BF02214657
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DOI: https://doi.org/10.1007/BF02214657