# Convergence (distribution)

$$X_1, X_2, X_3....$$ 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.

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}