Limit of $\frac 1 {\sqrt n} \int_0^{1} \int_0^{1} ...\int_0^{1} \sqrt {\sum_{i=1}^{n} x_i^{2}} dx_1...dx_n$

Question

What is limit of $\frac 1 {\sqrt n} \int_0^{1} \int_0^{1} ...\int_0^{1} \sqrt {\sum_{i=1}^{n} x_i^{2}} dx_1...dx_n$ as $n \to \infty$? This question was just deleted probably because there were errors in the statement. So I am posting it and providing my answer also.

Answer 1

Let $X_1,X_2,...$ be i.i.d. with uniform distribution on $(0,1)$. Then $ Y_n \to EX_1^{2}=\frac 1 3$ almost surely as $n \to \infty$ where $Y_n = \frac {X_1^{2}+X_2^{2}+...+{X_n^{2}}} n$ (by the strong law). Further $EY_n =\frac 1 3$ for each $n$. Hence $\{\sqrt Y_n\}$ converges almost surely and this sequence is uniformly integrable because the second moments are bounded. Hence $E\sqrt Y_n \to \frac 1 {\sqrt 3}$. This is exactly the limit we are asked to find!