$X_1, X_2, X_3....$ are independent random variables.
$P(X_n=0)=P(X_n=2)=1/4, P(X_n=-1)=1/2$.
Find the limit of:
$\frac{4\sqrt{n}(X_1+X_2+...+X_n)-7n}{n+(X_1+X_2+....+X_n)^2}$.
I computed: $EX_n=0, Var(X_n)=3/2$
I divided the nominator and denominator by $n^2$ and I tried to use the Central Limit Theorem and the Strong Law of Large Number.
Thanks in advance.
Let $S_n=X_1+\cdots +X_n$. Dividing both the numerator and the denominator by $n$, you want to study the convergence in distribution of the sequence \begin{align} \frac{4\frac{S_n}{\sqrt{n}}-7}{1+\left(\frac{S_n}{\sqrt{n}}\right)^2}. \end{align} Now we use the central limit theorem and the continuous mapping theorem: $(\frac{S_n}{\sqrt{n}})$ converges in distribution to a random variable $Y$ distributed as $N(0,3/2)$, and the function $\frac{4x-7}{1+x^2}$ is continuous. Therefore \begin{align} \frac{4\frac{S_n}{\sqrt{n}}-7}{1+\left(\frac{S_n}{\sqrt{n}}\right)^2} \to \frac{4Y-7}{1+Y^2} \end{align}