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I have this exercise

"Let: $$X_1,X_2, \ldots , Y_1,Y_2, \ldots \sim U (0,1)$$ independent and identically distributed random variables.

Is true that: $$\limsup \frac{X_n}{Y_n} = +\infty $$ almost sure?

I wanted to use the Borel Cantelli lemmas or the definition of $\limsup$, but I don't know how and if it's the right way....how can I do?

Thanks to all!

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You can use the second Borel-Cantelli lemma, which states: If $\{E_n\}$ are independent events with $\sum\Pr( E_n) = \infty$, then $\Pr(E_n \rm{,\,i.o.}) =1$.

For your exercise, try $E_n := \{ X_n / Y_n > k_n\}$, where ${k_n}$ is a sequence of positive reals tending to infinity. Argue that $\Pr(E_n) = 1/(2k_n)$ (draw a picture!), apply the lemma with an appropriate choice of $k_n$, and conclude that the event that $X_n/Y_n$ exceeds $k_n$ infinitely often has probability 1. Since the sequence $\{k_n\}$ tends to infinity, this last event implies that the sequence $X_n/Y_n$ has a limit superior of infinity, and you're done.

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  • $\begingroup$ how will we show that $\mathbb P(E_n)=\frac{1}{2k_n}$ $\endgroup$ – bunny Oct 17 '17 at 19:59

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