The following question is from the book [Quantitative Equity Portfolio Management: Modern Techniques and Applications] Page 47. I guess it could be solved using the Law of large numbers, but I'm not sure how to do it.
Given $L$ periods investment return $r_1,\cdots,r_L$, define geometric average as
$\displaystyle\mu_g=\Big(\prod_{i=1}^L(1+r_i)\Big)^{\frac{1}{L}}-1$
Suppose $r_i=\mu+\sigma\varepsilon_i$, where $\varepsilon_i$'s are independent standard normal random variables. Prove that as $L\to\infty,\;\;\; \mu_g\approx\mu-\frac{1}{2}\sigma^2$