Central Limit Theorem and convergence of transformed

Question

I have the following exercise to solve:

Let $X_n, n \geq1$ be a sequence of i.i.d random variables where each $X_n$ is a discrete random variable with distribution $P(X_n=1)=1-p$ and $P(X_n=2)=p$, with $0<p<1$. Determine the asymptotic behaviour for $n \rightarrow \infty$ of

$Z_n=n \frac{(\prod_{i=1}^{n}Xi)^{(1/n)}}{X_1^2+X_2^2+....X_n^2}$

I know I should apply the log transofrmation in the numerator but I get very confused after that.

Thank you in advance

Answer 1

Some steps:

1. Using the law of large numbers, we know that $n^{-1}\sum_{i=1}^nX_i^2\to \mathbb E\left[X_1^2\right]$ almost surely.
2. Since $\ln\left(\left(\prod_{i=1}^nX_i\right)^{1/n}\right)=\frac 1n\sum_{i=1}^n\ln X_i$, the law of large numbers applied to the i.i.d. sequence $\left(\ln X_i\right)_{i\geqslant 1}$ gives the asymptotic behavior of $\left(\prod_{i=1}^nX_i\right)^{1/n}$.