21 we often write this as plim n g w n g plim n w n

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We often write this as plim n g W n g plim n W n so we can pass the probability limit through continuous functions. Notice this contains the usual result on convergence for nonrandom sequences: lim n g a n g lim n a n if a n a and g  is continous at a . Slutsky’s Theorem has many applications in large-sample statistical analysis. 22
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EXAMPLES : Suppose W n 0 is such that plim W n c 0. Then (i) W n p c (ii) 1/ W n p 1/ c if c 0.If c 0, plim 1/ W n does not exist. These examples show that we can pass the probability limit through simple functions. By contrast, we cannot do the same with expected value: E W n E W n [in fact, E W n E W n by Jensen’s] E 1/ W n 1/ E W n [in fact, E 1/ W n 1/ E W n by Jensen’s] 23
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Slutsky’s theorem has many implications: plim a W n a plim W n which has, as a special case, plim aX n bY n a plim X n b plim Y n and then plim X n Y n plim X n plim Y n 24
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We can extend this to linear combinations that depend on n :if W n p W , a n a , and b n b , then a n W n b n p a W b A few more applications to nonlinear functions: plim X n Y n plim X n plim Y n plim X n / Y n plim X n /plim Y n if plim Y n 0. For the second result, it is possible that Y n 0 along the way, but if plim Y n 0 then P Y n 0 0. 25
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Slutsky’s Theorem applies to functions of matrices, too. For example, if Z n p A ,a k m matrix, and X n p b ,an m 1 vector, then Z n p A Z n Z n p A A Z n X n p A b 26
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If k m then tr Z n p tr A det Z n p det A because the trace and determinant functions are both continuous. Also, if det A 0 then A 1 exists ( A is nonsingular) and we can conclude Z n 1 p A 1 . If Z n is such that P Z n is singular 0as n ,wesay Z n : n 1,2,. .. is nonsingular with probability approaching one . It suffices that det Z n p det A 0. 27
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Consider a deterministic example. If Z n 30 0 n 50 / n then Z n converges to the nonsingular matrix A 01 . 28
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Even though Z n is singuar when n 50, it is nonsingular for all n 50, and so it makes sense to write Z n 1 A 1 1/3 0 01 as n . Generally, when Z n is random, we allow P Z n is singular) to be positive but converging to zero. What matters is that the probability limit of Z n is nonsingular.
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21 We often write this as plim n g W n g plim n W n so we...

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