I'm working through a proof of the Kelly Criterion found at http://www.elem.com/~btilly/kelly-criterion/. In checking up on some of the derivations I can't follow the last step in this sequence (equation below marked [*]):
(Note, $X_i$ is a random variable, E(X) is expectation of X)
$$ w_n=w_oX_1X_2...X_n\\ w_n=e^{log{(w_oX_1X_2...X_n)}}\\ w_n=e^{log(w_o) + log(X_1) + log(X_2) + ... + log(X_n)} [1]\\ $$
Using the law of large numbers: $ log(X_1) + log(X_2) + ... + log(X_n) = E(log(X))n + o(n)$
[o(n) is little-oh notation]
which reduces [1] to $$ w_n=e^{log(w_0) + E(log(X))n + o(n)}\\ w_n=e^{log(w_0) + E(log(X))n + o(1)n}\\ w_n=w_0 e^{(E(log(X)) + o(1))n}\\ w_n=w_0(e^{(E(log(X))}(1+ o(1)))^n [*]\\ $$
That last step [*] is where I get lost. Isn't that the equivalent of saying:
$$ e^{k + o(1)} = e^k(1+o(1)) $$
, where $k = E(log(X))$?
I don't see how to make that leap.
Any help appreciated.