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Assume $(X_{n})_{n\geq 1}$ is a sequence of random variables defined on the same probability space and taking values at $\ell_{2}$ (space of square-summable sequences). Next, assume that for some $a\in \ell_{2}$ $$ \sqrt{n}(X_{n} - a) \xrightarrow{a.s.} Z, $$ where $Z$ is a random variable.

Is this true that $$ X_{n} \xrightarrow{a.s.} a? $$

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Defining $$ Y_n= \sqrt n(X_n-a) - Z, $$ we have by hypothesis that $Y_n\xrightarrow{a.s.} 0$, so $$ X_n=\frac{Z + Y_n}{\sqrt n}+a \xrightarrow{n\to0} a. $$

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  • $\begingroup$ could you, please, clarify it... $\endgroup$ – LrM Nov 26 '20 at 14:19
  • $\begingroup$ Yes, but it would be nice to know exactly which part you failed to understand. $\endgroup$ – Ruy Nov 26 '20 at 14:21
  • $\begingroup$ I do not understand "By hypothesis $\sqrt n(X_n-a) = Z + Y_n$". $\endgroup$ – LrM Nov 26 '20 at 14:33
  • $\begingroup$ Note, I added that $Z$ is random. $\endgroup$ – LrM Nov 26 '20 at 14:42
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    $\begingroup$ Is it more clear now? $\endgroup$ – Ruy Nov 26 '20 at 15:27

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