Sequence of normalized random variables

[batman]

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]

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]

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]

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]

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.

 

[matheagle]

what if the X's blow up like

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

 

[CaptainBlack]

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 ..) ?