Limit of a sequence of random variables

Question

Suppose $Z_n$ is a sequence of independent random variables, which are uniformly picked from the interval $[1,2]$. Show that: $$ \lim_{n_\rightarrow \infty}P\left(\left|\sqrt[n] {Z_1 Z_2\cdots Z_n}-\frac{4}{e}\right|>\epsilon\right)=0. $$ I know I should probably apply the weak laws of large numbers, but I'm not sure how.

Answer 1

Hint: Apply the weak law to the log of $\sqrt[n]{Z_1Z_2\cdots Z_n}$:

$\ln\left[\left(Z_1Z_2\cdots Z_n\right)^{1/n}\right] =\frac1n \ln\left(Z_1Z_2\cdots Z_n\right) =\frac1n\sum_{i=1}^n\ln Z_i $. The resulting variables $\ln Z_1,\ln Z_2,\ldots,\ln Z_n$ are iid. To apply the LLN to them you need to calculate $E(\ln Z)=\int_1^2\ln z\,dz$.

Then use (or prove) the fact that if $X_n\to c$ in probability, with $c$ a constant, then $h(X_n)\to h(c)$ in probability whenever $h$ is continuous at $c$.

For your problem you need a function that will undo the effect of the previous log transformation: $h(x)=\exp (x)$ is the right choice.