Equivalently for all 0 there exists 0 b such that p w

This preview shows page 45 - 52 out of 88 pages.

Equivalently, for all 0 there exists 0 b such that P | W n | b 1 , n 1,2,... A common notation – more in Section 4 – is W n O p 1 (read “big oh pee one”). 45
Image of page 45

Subscribe to view the full document.

Some trivial examples of sequences bounded in probability are W n  n 1 / n Z and W n 1 n Z for a random variable Z . In the first case W n p Z and in the second case W n does not converge in probability. But it is bounded by | Z | . The definition of O p 1 applies element-by-element to vectors and matrices. 46
Image of page 46
Important Fact : If W n p W for a random variable W , then W n O p 1 . The simple example W n 1 n Z shows that the converse is not true: a random variable bounded in probability need not converge. Of course, a simple nonrandom example works, too: a n 1 n . (A bounded deterministic sequence does not necessarily converge.) 47
Image of page 47

Subscribe to view the full document.

EXAMPLE : Let X i : i 1,2,... be pairwise uncorrelated random variables with finite second moment, mean zero, and Var X i E X i 2 2 . The variance of the partial sum Y n X 1 X 2 ... X n is n 2 , which increases as a linear function of n . It seems impossible that Y n could be O p 1 . [As an exercise, show that if the X i are all Normal 0,1 then, for any 0 b , P | Y n | b 0 as n , which clearly violates the definition of O p 1 .] 48
Image of page 48
But if we properly standardize the sum, we get a O p 1 random variable. In particular, if W n X 1 X 2 ... X n / n then Var W n 2 . We can use Chebyshev’s inequality to verify that W n O p 1 : P | W n | b Var W n b 2 2 b 2 Therefore, in the definition of O p 1 , for any 0 choose b / and then P | W n | b . 49
Image of page 49

Subscribe to view the full document.

EXERCISE: What happens if E X i for 0? SOLUTION: If E X i then E W n n . It is pretty clear that for 0, W n is not bounded in probability. In fact, we can write W n n Z 1 Z 2 ... Z n / n where Z j X j j , and so W n n O p 1 , which is clearly an unbounded sequence. 50
Image of page 50
Not all unbounded sequences are the same. For example, the sequence  1 n n alternates between negative and positive values, with its absolute value . But other sequences continue to grow (or shrink), such as n or 2 n . It is useful to have the notion of W n p . This is just shorthand for the following: for all 0 b , P W n b 1. Similarly, W n p − means P W n b 1 for all 0 b .
Image of page 51

Subscribe to view the full document.

Image of page 52
  • Fall '12
  • Jeff
  • Probability, Probability theory, Convergence, WLLN

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

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