Sequence of normalized random variables

batman

New member
Let X_1, X_2, ... be a sequence of random variables and define Y_i = X_i/E[X_i]. Does the sequence Y_1, Y_2, ... always convergence to a random variable with mean 1?

CaptainBlack

Well-known member
Let X_1, X_2, ... be a sequence of random variables and define Y_i = X_i/E[X_i]. Does the sequence Y_1, Y_2, ... always convergence to a random variable with mean 1?
There is something missing from your statement of the problem, the $$Y_i$$s all have mean 1, but there is no reason why they should converge without some further restriction on the $$X$$s.

CB

batman

New member
So what should be the restriction on the X_i's? In particular, can the Y_i's still convergence if the X_i's go to infinity?

CaptainBlack

Well-known member
So what should be the restriction on the X_i's? In particular, can the Y_i's still convergence if the X_i's go to infinity?
It would be better if you provide the context and/or further background for your question.

CB

batman

New member
There is no context and/or further background. It was just a question that came to my mind while studying convergence of random variables. I just thought a normalized sequence would always have mean 1 in the limit and I was wondering whether there would be a general condition when it converges.

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matheagle

New member
what if the X's blow up like

$P(X_n=2^n)=2^{-n}$ and $P(X_n=0)=1-2^{-n}$

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CaptainBlack

Well-known member
There is no context and/or further background. It was just a question that came to my mind while studying convergence of random variables. I just thought a normalized sequence would always have mean 1 in the limit and I was wondering whether there would be a general condition when it converges.
Suppose $$X_{2k}\sim N(1,1)$$ and $$X_{2k-1} \sim U(0,2),\ k=1, 2, ..$$ Now does the sequence $$\{X_i\}$$ converge (in whatever sense ..) ?