$X_1, X_2, X_3....$ are independent random variables.

$P(X_n=0)=P(X_n=2)=1/4, P(X_n=-1)=1/2$.

Find the limit of:


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.

  • $\begingroup$ en.wikipedia.org/wiki/Slutsky%27s_theorem Theorem would have helped if the denominator didn't have square of sum. In that case, denominator would have converged to the constant 1 after dividing by 1 and numerator to an appropriate Gaussian. $\endgroup$ – Ankitp Feb 20 at 17:18

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}


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