In something I'm working on, the expression $$\lim_{n\rightarrow\infty}\exp\left(-\frac{1}{n}\log\left(\prod_{i=1}^{n}X_{i}+\prod_{i=1}^{n}Y_{i}\right)\right)$$ came up, in which all $X_{i}$ $iid$ and all $Y_{i}$ $iid$. Is there any way to simplify or rewrite this nicely, hopefully with respect to expectations/central moments?
For example, in another location, the expression $$\lim_{n\rightarrow\infty}\exp\left(-\frac{1}{n}\log\left(\prod_{i=1}^{n}X_{i}\right)\right)$$ came up, and I was able to use the properties of log and the law of large numbers to get $$\exp\left(E[\log(X)]\right)$$ Then I substituted in $E[X]+\delta$ for $X$, and used a Taylor expansion, which can be truncated to get $$\exp\left(-\frac{\sigma^2_{X}}{2\mu^{2}_{X}}\right)$$ This is the kind of thing I'm hoping can be done with the first expression listed in this question.