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kingwinner
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I was reading some proofs about the convergence of random variables, and here are the little bits that I couldn't figure out...
1) Let Xn be a sequence of random variables, and let Xnk be a subsequence of it. If Xn conveges in probability to X, then Xnk also conveges in probability to X. WHY?
2) I was looking at a theorem: if E(Y)<∞, then Y<∞ almost surely. Now I am puzzled by the notation. What does it MEAN to say that Y=∞ or Y<∞?
For example, if Y is a Poisson random variable, then the possible values are 0,1,2,..., (there is no upper bound). Is it true to say that Y=∞ in this case?
3) If Xn4 converges to 0 almost surely, then is it true to say that Xn also converges to 0 almost surely? Why or why not?
4) The moment generating function(mgf) determines the distribution uniquely, so we can use mgf to find the distributions of random varibles. If the mgf already does the job, what is the point of introducing the "characteristic function"?
Can someone please explain?
Any help is much appreciated! :)
1) Let Xn be a sequence of random variables, and let Xnk be a subsequence of it. If Xn conveges in probability to X, then Xnk also conveges in probability to X. WHY?
2) I was looking at a theorem: if E(Y)<∞, then Y<∞ almost surely. Now I am puzzled by the notation. What does it MEAN to say that Y=∞ or Y<∞?
For example, if Y is a Poisson random variable, then the possible values are 0,1,2,..., (there is no upper bound). Is it true to say that Y=∞ in this case?
3) If Xn4 converges to 0 almost surely, then is it true to say that Xn also converges to 0 almost surely? Why or why not?
4) The moment generating function(mgf) determines the distribution uniquely, so we can use mgf to find the distributions of random varibles. If the mgf already does the job, what is the point of introducing the "characteristic function"?
Can someone please explain?
Any help is much appreciated! :)