# a.s. convergence in $\ell_2$ space

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 $$X_{n} \xrightarrow{a.s.} a?$$

Defining $$Y_n= \sqrt n(X_n-a) - Z,$$ we have by hypothesis that $$Y_n\xrightarrow{a.s.} 0$$, so $$X_n=\frac{Z + Y_n}{\sqrt n}+a \xrightarrow{n\to0} a.$$
• I do not understand "By hypothesis $\sqrt n(X_n-a) = Z + Y_n$". – LrM Nov 26 '20 at 14:33
• Note, I added that $Z$ is random. – LrM Nov 26 '20 at 14:42