Hey I want to see if I have done everything right proving this statements:
Let $(X_n)n∈N$ independent, identically distributed random variables with $X_i ∼ N (0, 1)$. Be also $Z ∼ N(0, 1)$. For all $n ∈ N$ let be defined:
$Y_n:= \frac{X_1+...+X_n}{n}$ and $Z_n:= \frac{X_1+...+X_n}{\sqrt{n}}$
I have to prove that $Y_n \overset{D}{\rightarrow}0$ and $Z_n \overset{D}{\rightarrow}Z$.
For the first statement I have done this:
In the Script we had that if $Y_n$ converges in distributions towards a constant $c$ than $Y_n \overset{D}{\rightarrow} c$
Than $P(|X_1+...+X_n|>zn)\leq \frac{Var(X_1+...+X_n)}{n^2z^2}=\frac{n}{n^2z^2}=\frac{1}{nz^2}\overset{\infty}{\rightarrow}=0$
For the second statement I thought doing like this: $\lim_{n\rightarrow\infty}P(X_1+...+X_n\leq z\sqrt{n})=P(\frac{(X_1+...+X_n)-nµ}{\sqrt{nσ^2}}\leq\frac{z\sqrt{n}-nµ}{\sqrt{nσ^2}}=z)=\Phi(z)$
Is everything right? Have I done any errors?
For the convergence of $Y_n$, you have to specify that $z$ is fixed and positive, and also justify that the variance of the sum $X_1+\dots+X_n$ is the sum of variances.
For the second part, you cannot have that that a limit as $n$ goes to infinity equal something depending on $n$. Also, you have to justify that $(X_1+\dots+X_n)/\sqrt n$ has a standard normal distribution.