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$(X_n), (Y_n)$ independent sequences of iid random variables such that
$EX_1=EY_1=0$,
$\operatorname{Var}(X_1)=\operatorname{Var}(X_2)=\sigma^2$.

$S_n=\sum_{k=1}^nX_k$, $T_n=\sum_{k=1}^nY_k$.

1)
Show that $(\frac{S_n}{\sqrt{n}}, \frac{T_n}{\sqrt{n}})$ converges in distribution and find the limit distribution.

Let $Z_k=(X_k,Y_k), \sum_{k=1}^nZ_k=(S_n, T_n)$

From CLT I have:
$\frac{\sum_{k=1}^nZ_k}{\sqrt{n}}=(\frac{S_n}{\sqrt{n}}, \frac{T_n}{\sqrt{n}})$ converges to a random variable with normal distribution $N_2(0,K)$, where $K=\begin{bmatrix} \sigma^2&0 \\0&\sigma^2 \end{bmatrix}$
Is this okay?

2)
Show that $P(\sqrt{S_n^2+T_n^2}>\sqrt{n})$ has a limit.
I tried this way (using Chebyshev's inequality):
$$P(\sqrt{S_n^2+T_n^2}>\sqrt{n})=P(\sqrt{\frac{S_n^2+T_n^2}{n}}>1) \le \frac{1}{\sqrt{n}}E\sqrt{2S_n^2}=\sqrt{\frac{2}{n}}E|S_n|$$
but because of the absolute value I don't know if it is zero, or do I?

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For 1), the result is correct. Depend on what you already know and are supposed to use. For example, if you only have the one dimensional result at your disposal, then you have to justify that $\left(aS_n/\sqrt n+bT_n/\sqrt n\right)_{n\geqslant 1}$ converges in distribution to a centered normal distribution with variance $\sigma^2\left(a+b\right)^2$ for all real numbers $a$ and $b$ (Cramer-Wold).

For 2), let $F\colon \mathbb R^2\to\mathbb R$ be defined by $F\left(x,y\right)=\sqrt{x^2+y^2}$. Then the answer follows from the two following sub-questions:

  • i) Which interesting property with respect to the convergence in distribution has the function $F$?
  • ii) What is $F\left(S_n/\sqrt n,T_n/\sqrt n\right)$?
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  • $\begingroup$ i) Maybe $X_n \to X, Y_n \to Y \implies F(X_n,Y_n) \to F(X,Y)$? ii) $F\left(S_n/\sqrt n,T_n/\sqrt n\right)=\sqrt{\frac{S_n^2+t_n^2}{n}}$ So I will have to find (using i) and ii)) limit of distributions of $\sqrt{\frac{S_n^2+t_n^2}{n}}$ in point $1$ and I will get limit of $P(\sqrt{S_n^2+T_n^2}>\sqrt{n})$? $\endgroup$ – Michalina Feb 5 '18 at 15:12
  • $\begingroup$ For i), we need the convergence of $(X_n,Y_n)\to (X,Y)$ (not only of the marginales) and fortunately we have it. ii) Yes. $\endgroup$ – Davide Giraudo Feb 5 '18 at 15:18
  • $\begingroup$ Okay, so I know that $(X,Y)$ has density $N_2(0,K)$, now I have to find the density of $\sqrt{X^2+Y^2}$, but I don't know how $\endgroup$ – Michalina Feb 5 '18 at 23:58
  • $\begingroup$ You want to compute $\Pr(\sqrt{X^2+Y^2} \in B)$ for a Borel subset $B$ of the real line. Express this as $\int_{\mathbb R^2}\mathbf 1_{\sqrt{x^2+y^2}\in B}g(x,y)dxdy$, where $g$ is a density of $N_2(0,K)$. Then use polar coordinates. $\endgroup$ – Davide Giraudo Feb 6 '18 at 9:35

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