My Math Forum I have this homework that is driving me crazy

 December 24th, 2011, 09:49 AM #1 Newbie   Joined: Dec 2011 Posts: 8 Thanks: 0 I have this homework that is driving me crazy Hi all Let X1,X2,…,Xn be iid(independent identical distribution).let ,let Yn=sign(?Xi),(i.e. the product of Xi ,i from 1 to n) where by Sign(a)={-1 if a<0 1 if a>0 Study the sequence Yn in details in terms of distributional properties, convergence in probability of Yn and ?Yn/n, independence of Ys and also the Ys and the Xs,etc. hope you could help me with any ideas to solve this homework
 December 24th, 2011, 12:24 PM #2 Newbie   Joined: Dec 2011 Posts: 1 Thanks: 0 Re: I have this homework that is driving me crazy Hi all Let X1,X2,…,Xn be iid (independent identical distribution)normal distribution (0,1).let Yn=sign(?Xi),(i.e. the product of Xi ,i from 1 to n) where by Sign(a)={-1 if a<0 1 if a>0 Study the sequence Yn in details in terms of distributional properties, convergence in probability of Yn and ?Yn/n, independence of Ys and also the Ys and the Xs,etc. hope you could help me with any ideas to solve this homework
 December 24th, 2011, 02:12 PM #3 Global Moderator   Joined: May 2007 Posts: 6,660 Thanks: 647 Re: I have this homework that is driving me crazy You can start by noting that for for each Xi only the sign matters, so you can simplify by assuming Xi has two values +1 and -1, each with probability 1/2. Then Yn can be studied more easily. Y1 can be +1 or -1 with probability each 1/2. For each Yn note that Yn = XnY[n-1]. Since Xn is independent of previous X's, the net result is that Yn is either the same sign or opposite sign of Y[n-1] each probability 1/2. Pu it all together and you have a sequence Yn of independent random variable, each one of which has the same distribution. (+1 or -1, probs = 1/2). You should be able to do the rest yourself.

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