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Showing posts from March, 2015

Probability Theory: Convergence in Distribution Problem

Let's solve some theoretical problem in probability, specifically on convergence. The problem below is originally from Exercise 5.42 of Casella and Berger (2001). And I just want to share my solution on this. If there is an incorrect argument below, I would be happy if you could point that to me. Problem Let X_1, X_2,\cdots be iid (independent and identically distributed) and X_{(n)}=\max_{1\leq i\leq n}x_i. If x_i\sim beta(1,\beta), find a value of \nu so that n^{\nu}(1-X_{(n)}) converges in distribution; If x_i\sim exponential(1), find a sequence a_n so that X_{(n)}-a_n converges in distribution. Solution Let Y_n=n^{\nu}(1-X_{(n)}), we say that Y_n\rightarrow Y in distribution. If \lim_{n\rightarrow \infty}F_{Y_n}(y)=F_Y(y).
Then, $$ \begin{aligned} \lim_{n\rightarrow\infty}F_{Y_n}(y)&=\lim_{n\rightarrow\infty}P(Y_n\leq y)=\lim_{n\rightarrow\infty}P(n^{\nu}(1-X_{(n)})\leq y)\\ &=\lim_{n\rightarrow\infty}P\left(1-X_{(n)}\leq \frac{y}{n^{...