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Sheet12

# Sheet12 - (12.4 Suppose X n has a Bin n p n distribution...

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Statistics 330/600 Due: Thursday 21 April 2005: Sheet 12 *(12.1) Let X be a metric space. (i) UGMTP Problem 7.7 (ii) Let P be a probability measure on B ( X ) . Suppose { X n : n N } is a sequence of random elements of X , with the following property: for every subsequence { X n i : i N } there exists a sub-subsequence { X n i j : j N } for which X n i j Ã P as j →∞ . Show that X n Ã P . *(12.2) (cid linear) UGMTP Problem 7.13, but only the case for random vectors. Hint: What do you know about the “random vector” ( X n , A n , B n ) ? *(12.3) (cid via Fubini) UGMTP Problem 7.10.
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Unformatted text preview: (12.4) Suppose X n has a Bin ( n , p n ) distribution for which lim inf n np n ( 1 − p n ) < ∞ . Show that ( X n − np n )/ √ np n ( 1 − p n ) does not converge in distribution to N ( , 1 ) . *(12.5) Suppose X n has a Poisson ( n ) distribution. Show that √ X n − √ n Ã N ( , 1 / 4 ) . Hint: Read UGMTP Example 7.14 then consider the random vector ( ( X n − n )/ √ n , √ X n / n ) ....
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