sec05 - Large Sample Theory Ferguson Exercises, Section 5,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Large Sample Theory Ferguson Exercises, Section 5, Central Limit Theorems. 1. (a) Using a Chebyshev’s-Inequality-like argument, show that (assuming the expec- tations exist) E | X | 2+ α t α E[ X 2 I( | X |≥ t )] for all α> 0and t> 0. (b) Using part (a) and Lindeberg, prove Liapounov’s Theorem: Let X n 1 ,X n 2 ,...,X nn be independent with E X nj =0andE | X nj | 2+ α < for some 0anda l l n and j .L e t Z n = n j =1 X nj and B 2 n =Var Z n = n j =1 Var X nj .T h e n Z n /B n L −→ N (0 , 1), provided 1 B 2+ α n n X j =1 E | X nj | 2+ α 0as n →∞ . 2. Let X 1 2 ,... be independent exponential random variables with means β 1 2 respectively, and let Z n = X 1 + ··· + X n . Show that if max 1 j n β 2 j / n j =1 β 2 j 0a s n ,then( Z n E Z n ) / Z n L (0 , 1). (Use Liapounov’s Theorem with α =2.) 3. (a) Let X 1 2 be independent Poisson random variables with means λ 1 2 respectively, and let Z n = X 1 + + X n . Show that ( Z n E Z n ) / Z n L (0 , 1) if and only if n 1 λ j . (b) Show that this can provide an example to show you can get asymptotic normality without the Lindeberg condition being satis±ed. 4. As an illustration of the use of Kendall’s tau, here is a famous little example taken from M. G. Kendall’s 1948 book, Rank Correlation Methods . Suppose a number of boys are ranked according to their ability in mathematics and music. Such a pair of rankings
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Spring '09 term at Aarhus Universitet.

Page1 / 3

sec05 - Large Sample Theory Ferguson Exercises, Section 5,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online