{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This 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 > 0 and 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 = 0 and E | X nj | 2+ α < for some α > 0 and all n and j . Let Z n = n j =1 X nj and B 2 n = Var Z n = n j =1 Var X nj . Then Z n /B n L −→ N (0 , 1), provided 1 B 2+ α n n j =1 E | X nj | 2+ α 0 as n → ∞ . 2. Let X 1 , X 2 , . . . be independent exponential random variables with means β 1 , β 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 0 as n → ∞ , then ( Z n E Z n ) / Var Z n L −→ N (0 , 1). (Use Liapounov’s Theorem with α = 2.) 3. (a) Let X 1 , X 2 , . . . be independent Poisson random variables with means λ 1 , λ 2 , . . . respectively, and let Z n = X 1 + · · · + X n . Show that ( Z n E Z n ) / Var Z n L −→ N (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 satisfied. 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
Image of page 1

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

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

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern