slides_7_converge

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

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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

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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
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

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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
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

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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
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 .

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• Fall '12
• Jeff
• Probability, Probability theory, Convergence, WLLN

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