Let's say I have some hypothetical investment which can earn me some return $i$, where $i$ is a random variable. After $n$ periods, my expected return on my investment is:
$$ E[((1+i_1)(1+i_2)(1+i_3)\cdots(1+i_n))^{1/n}] $$
$$ =E[(1+i_1)^{1/n}(1+i_2)^{1/n}(1+i_3)^{1/n}\cdots(1+i_n)^{1/n}] $$
Since the returns are all independent of each other (we will assume), then we can simplify:
$$ =E[(1+i)^{1/n}]^{n} $$ Since the expectation has $n$ embedded in it, it will depend on the value of $n$. My question is, what does this expression look like as $n$ goes to infinity: $$ \lim_{n\to \infty} E[(1+i)^{1/n}]^{n} $$
I suspect that the answer is the Geometric Mean of $1+i$: $$ e^{E[ln(1+i)]} $$
Does anyone have any thoughts on how to prove this?