Let $X_1$,$X_2$,...,$X_n$ be i.i.d. random variables with common density $$f(x)=\frac{\alpha}{x^{\alpha+1}}, $$ for $\alpha >0$ and $x>1$. Define $S_n=[\prod_{i=1}^{n}X_i]^{1/n}$.
(a) Use the CDF technique to show that $\ln(X_i)$ follows an exponential distribution and give the mean and variance.
(b) Show that $E(\ln(S_n))=E(\ln(X_1))=\frac{1}{\alpha}$.
(c) Use the Central Limit Theorem to find, as $n \to \infty$, the limiting distribution of $$\sqrt{n}[\ln(S_n)-\alpha^{-1}].$$
(d) Suppose that $\alpha=10$ and $n=100$. Find $P(S_n>1.12)$. Give the result as a function of the cumulative c.d.f. of the standard normal distribution $\phi(\circ)$.
I've done (a) and (b) but am stuck on (c) and consequently cannot do (d). I know I can put the $\sqrt{n}$ in the denominator as $1/\sqrt{n}$ but from there I'm stuck. Any help would be greatly appreciated.