For nonrandom sequences we drop the p subscript a n o

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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 0. We started with the premise n a n o p 1 . 59
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We have similar definitions for being bounded in probability. If n W n is bounded in probability we write W n O p n which is read “ W n is of order no greater than n in probability.” The definition applies to nonrandom sequences. The notation O p 1 again comes with 0. The definitions imply that if W n o p n then W n O p n . In practice we need relatively little of the o p / O p apparatus. It is useful for theoretical work, but we will mainly use o p 1 , O p 1 , O p n 1/2 , and O p n 1/2 . 60
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EXAMPLE : Earlier we considered a standardized partial sum. We can appply the O p apparatus to the partial sum and sample average. Let X i : i 1,2,... be random variables with finite second moment that are pairwise uncorrelated, and assume E X i 0 and Var X i E X i 2 2 . Let Y n be the partial sum and X ̄ n Y n / n be the sample average. Using Chebyshev’s inequality we showed that n 1/2 Y n O p 1 which means Y n O p n 1/2 . We can also write this as n 1/2 X ̄ n O p 1 ,which means X ̄ n O p n 1/2 . 61
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Note that when Y n is the partial sum, we can also write Y n o p n 1/2 for any 0, that is, n 1/2 Y n p 0. On the other hand, Y n O p n 1/2 for and 0. In other words, n 1/2 is the function of n that we must divide by in order to obtain a sequence that is bounded but does not have a variance shrinking to zero. We will make a stronger statement about such sequences in the next section. If E X i 0, we need to use n 1/2 X ̄ n n 1/2 Y n n as the sequence that is O p 1 . 62
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6 . Convergence in Distribution Convergence in distribution is important for approximating probabilities involving averages and functions of them. It is very important for approximating the sampling distributions of estimators in a variety of settings (later). Consider the following general problem. We assume X i : i 1,2,... is a sequence of i.i.d. random variables with finite second moment. Let E X i , 2 Var X i . Let X ̄ n be the sample average of the first n variables in the sequence.
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