Let (X₁, Y₁),⋯, (XN, YN) be i.i.d. rv's where the X's are nonnegative integer-valued. Conditional on σ Xₖ the asymptotic distribution of σ Yₖ and σ aₖ Xₖ are derived by general methods. Some applications are briefly discussed: sampling without replacement, the classical occupancy problem, the Wilcoxon statistic, the Poisson index of dispersion, testing geometric versus Poisson distribution.
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Lars Holst (1979) studied this question.