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
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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
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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
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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
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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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