On the distribution of the leading statistics for the bounded deviated permutations

Abstract

本篇論文的研究目的是要在均勻分佈的假設下，探索bounded deviated permutations的第一位置統計量的分佈情形，而我們猜測其將為常態分佈。 定義好與第一位置統計量相關的隨機變數後，藉由考慮雙變數生成函數，我們可以計算此隨機變數的平均數和變異數，本篇論文將把這個方法應用在三個特殊的情形上。因為這個雙變數生成函數的係數並沒有closed form，在計算過程中，我們會使用Hayman's formula求其漸進式。最後，使用電腦計算，這三個特殊的情形確實收斂到常態分佈，證實了我們的猜測。The purpose of the thesis is to investigate the distribution of the leading statistic in the bounded deviated permutations S_{n+1}^{ℓ,r}, assuming the uniform distribution in S_{n+1}^{ℓ,r}. Define the random variable X_{n} to take the value k if π₁=k+1 for π=π₁π₂⋯π_{n+1}∈S_{n+1}^{ℓ,r}. By considering the bivariate generating function A(z,u), we could calculate the expected value and the standard deviation for X_{n}. The method is then applied to three specific cases, S_{n+1}^{1,2}, S_{n+1}^{1,3} and S_{n+1}^{2,2}. Since the coefficients λ_{n,k} of the bivariate generating function do not have a closed form, we will apply the Hayman method to get its asymptotic formula. Finally, by running computer programs, the convergence of the normal distribution on these three cases are verified.