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Abstract
This dissertation discusses several statistical results under multinomial distribution with infinite categories. Firstly, the discussion focuses on Simpson's diversity index and Turing's formula. We established an unbiased estimate for the newly proposed Generalized Simpson's indices and the associated asymptotic properties and showed that the parameters of a multinomial distribution may be re-parameterized as a set of Generalized Simpson's diversity indices. Secondly, two-dimensional asymptotic normality of a non-parametric sample coverage estimate based on Turing's formulae was derived under a fixed underlying probability distribution {p_k; k = 1, 2, · · · } where all p_k > 0. Thirdly, the dissertation also establishes a previously unknown sufficient condition for the second order Turing's formula. The newly derived asymptotic results based on Turing's formula paves a possible way to establish a new estimating approach for Hill's tail probability model.