iid uniform

May 2008
50
1
Assume that \(\displaystyle X_1,X_2,...,X_n\) are iid uniform on [0,1]. Show that \(\displaystyle (X_1X_2...X_n)^\frac{1}{n}\to e^{-1}\) a.e. for \(\displaystyle n\to\infty\).(Itwasntme)

I appreciate any help.
 

Laurent

MHF Hall of Honor
Aug 2008
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Paris, France
Assume that \(\displaystyle X_1,X_2,...,X_n\) are iid uniform on [0,1]. Show that \(\displaystyle (X_1X_2...X_n)^\frac{1}{n}\to e^{-1}\) a.e. for \(\displaystyle n\to\infty\).(Itwasntme)

I appreciate any help.
Express the power using \(\displaystyle a^b=e^{b\log a}\) and use the law of large numbers.

(Or equivalently take the logarithm of the sequence and apply the law of large numbers)
 
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matheagle

MHF Hall of Honor
Feb 2009
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you also need to show that

\(\displaystyle E(\ln X)=\int_0^1 \ln xdx=-1\)
 
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May 2008
50
1
Thanks. That I was able to prove.(Clapping)