qtbabe9876a9

2022-05-29

Suppose we have a sequence of positive random variables ${X}_{1},{X}_{2},...,X$. I am trying to prove a characterization of almost sure convergence.
It states that ${X}_{n}\to X$ almost surely iff for every $ϵ>0$, $\underset{n\to \mathrm{\infty }}{lim}P\left[\underset{k\ge n}{sup}\frac{{X}_{k}}{X}>1+ϵ\right]=0$ and $\underset{n\to \mathrm{\infty }}{lim}P\left[\underset{k\ge n}{sup}\frac{X}{{X}_{k}}>1+ϵ\right]=0$.
If I assume almost sure convergence, then the implication is easy but I am not being able to prove the other way round.

concludirgt

Expert

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Since all the random variables are positive,

Thus,

But

That is, $\mathsf{P}\left(lim sup{X}_{n}/X>1\right)=0$ is equivalent to

and, similarly, $\mathsf{P}\left(lim inf{X}_{n}/X<1\right)=0$ is equivalent to

The last condition is equivalent to $\underset{n\to \mathrm{\infty }}{lim}\mathsf{P}\phantom{\rule{negativethinmathspace}{0ex}}\left(\underset{k\ge n}{sup}X/{X}_{k}\ge 1+ϵ\right)=0\phantom{\rule{1em}{0ex}}\mathrm{\forall }ϵ>0$.

Trevor Wood

Expert

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let

Then ${A}_{n+1}^{ϵ}\subseteq {A}_{n}^{ϵ}$ and ${B}_{n+1}^{ϵ}\subseteq {B}_{n}^{ϵ}$, thus

and

Since the above hold for every $ϵ>0$ we conclude that

These two are equivalent to the following two

Now observe that $\underset{n\to \mathrm{\infty }}{lim sup}\frac{{X}_{n}}{X}=\frac{1}{X}\cdot \underset{n\to \mathrm{\infty }}{lim sup}{X}_{n}$ and $\underset{n\to \mathrm{\infty }}{lim sup}\frac{X}{{X}_{n}}=X\cdot \underset{n\to \mathrm{\infty }}{lim sup}\frac{1}{{X}_{n}}=X\cdot \frac{1}{\underset{n\to \mathrm{\infty }}{lim inf}{X}_{n}}.$. Therefore,

taking intersections we conclude that ${X}_{n}\stackrel{\text{a.s}}{⟶}X$

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