slides_7_converge

# Because n log b n thus it makes sense to write w n p

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because n log b / n . Thus, it makes sense to write W n p . 52

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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
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

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5 . More on o p and O p The algebra of o p and O p is very important for asymptotic analysis. Earlier we noted that if W n p 0 we sometimes write W n o p 1 .If W n is bounded in probability, we write W n O p 1 . We summarize the important properties of o p 1 and O p 1 , which can be proven by using the definitions with some calculations. 55
Properties of o p 1 and O p 1 (i) If W n p W for any random variable W (including a constant) then W n O p 1 . (ii) It immediately follows from (i) that if W n o p 1 then W n O p 1 . (iii) If W n O p 1 and Z n O p 1 then W n Z n O p 1 .A convenient way to express this is O p 1 O p 1 O p 1 It follows immediately that o p 1 O p 1 O p 1 56

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(iv) o p 1 o p 1 o p 1 (v) O p 1  O p 1 O p 1 (vi) o p 1 O p 1 o p 1 , from which it follows o p 1 o p 1 o p 1 These facts are used regularly in large-sample analysis of estimators and test statistics. Fact (vi) is crucial for obtaining approximate sampling distributions (later). These relationships hold when we add and multiply vectors and matrices, too. 57
Where does the o p 1 , O p 1 notation come from? It comes from more general ways of describing “how big” random sequences are. If W n is a random sequence and n W n p 0 then we write W n o p n , which is read “ W n is of order less than n in probability.” 58

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Note that the definition applies to any , and it is useful for negative and positive values. For nonrandom sequences, we drop the “ p ” subscript: a n o n if n a n 0 and we say “ a n is of order less than n .” We get o p 1 when 0 in the definition: n 0 1. If then W n o p n  W n o p n , as is easily seen by n W n n n n W n n n W n o 1  o p 1 o p 1 , because n 0 when
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because n log b n Thus it makes sense to write W n p 52 For...

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