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# ps6 - Problem Set 6 ECON 837 Prof Simon Woodcock Spring...

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Unformatted text preview: Problem Set 6 ECON 837 Prof. Simon Woodcock, Spring 2006 Due: March 17 in class 1. Suppose that plim X n = 2 and plim Y n = 3 : Find plim X n Y n ; plim ( X n + Y n ) ; plim ( X 2 n + Y 2 n ) : 2. (Casella and Berger, 1990) Let X 1 ; X 2 ; ::: be a random sequence that converges in probability to a constant a: Assume that Pr [ X i &amp;gt; 0] = 1 for all i: (a) Without using the Slutsky Theorem , show that the random sequence Y 1 ; Y 2 ; ::: de&amp;amp;ned by Y i = p X i converges in probability to p a: (b) Show that if a &amp;gt; ; the sequence Y 1 ; Y 2 ; ::: de&amp;amp;ned by Y i = a=X i converges in probability to 1. 3. Suppose that X n = 3 &amp;amp; 1 n 2 Y n = p n &amp;amp; Z n &amp;amp; where &amp;amp; Z n = 1 n P n i =1 Z i ; and the Z i are iid with mean zero and variance &amp;amp; 2 : Find the limiting distributions of (a) X n + Y n (b) X n Y n (c) Y 2 n : 4. [Final Exam, 2004] Suppose f x i ; y i g n i =1 are an iid random sample from some distribution with at least 4 moments. Denote E [ x ] by x and E [ y ] by y : Let = x &amp;amp; y and let...
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ps6 - Problem Set 6 ECON 837 Prof Simon Woodcock Spring...

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