slides_7_converge

In the previous exercise with 0 w n p 51 example

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In the previous exercise with 0, W n p . 51

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EXAMPLE : Suppose X i : i 1,2,... are i.i.d. Normal 1,1 random variables, and define W n exp i 1 n X i . Then W n has the Lognormal n , n distribution. It follows that, for any b 0, P W n b 1 P W n b 1 log b n n n log b / n 1 as n because n log b / n . Thus, it makes sense to write W n p . 52
For a certain class of O p 1 sequences, taking continuous transformations is guaranteed to preserve the O p 1 property. We will cover these sequences in Section 6. We have to be careful in general. For example, if W n X / n where X 0, W n is O p 1 (in fact, W n p 0) but log W n log X log n is not O p 1 . 53

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Taking linear transformations is never a problem. As an exercise, you can show that if W nj : n 1,2,... are O p 1 for j 1,..., k , then so is Y n a 1 W n 1 ... a k W nk c for any constants a 1 ,..., a k , c . 54