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- #1

- Thread starter batman
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- Thread starter
- #1

- Jan 26, 2012

- 890

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.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?

CB

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- #3

- Jan 26, 2012

- 890

It would be better if you provide the context and/or further background for your question.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?

CB

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- #5

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.

Last edited:

- Jan 26, 2012

- 890

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 ..) ?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.