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December 24th, 2011, 08:49 AM   #1
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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 Y`s and also the Y`s and the X`s,etc.
hope you could help me with any ideas to solve this homework
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December 24th, 2011, 11:24 AM   #2
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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 Y`s and also the Y`s and the X`s,etc.
hope you could help me with any ideas to solve this homework
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December 24th, 2011, 01:12 PM   #3
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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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