Description
In a game, you can bet on the outputs 1, ... . , m, which are drawn with the
probabilities p1, p2, ... , pm.
If output i is drawn, the stakes on i are paid back m-fold, all the other stakes are lost.
A player plays according to the following strategy:
he distributes his capital K in the ratio q1 : q2 : ... : qm
among the possible outcomes, and uses the profit from one round as a stake in the next.
Find following limit:
$\lim_{x\to\infty} \frac{1}{n}*log(K_n)$
My Attempt
Transform the equation until the law of large numbers can be applied
$X_n = \frac{K_n}{K_{n-1}}$
$K_n = K_0*X_1*X_2*...*X_n$
$\lim_{x\to\infty} \frac{1}{n}*log(K_n)$
$= \lim_{x\to\infty} \frac{1}{n}*log(K_0*X_1*X_2*...*X_n)$
$= \lim_{x\to\infty} \frac{1}{n}*(log(K_0) + \sum_{i=1}^{n}log(X_i))$
I guess this is the place where I should use the law of large numbers
to get $\mathbb{E}(log(X))$, but I don't know how.
Could someone help me out, or at least give me a hint?
At the $n$-th stage of the game, the player holds $K_n$, a random integer $i_n\in\{1,\dots,m\}$ is chosen with probability $p_1,\dots, p_m$, in which case the player's new wealth is $K_{n+1} = mq_{i_n}K_n$. As you suggested, we write $$\log K_n - n\log m = \sum_{t=1}^n \log q_{i_t} $$ which is a sum of iid random variables with mean $\sum_{i=1}^m p_i\log q_i$. Thanks to the law of large numbers, $$\lim_{n\to\infty}\frac{\log K_n}{n} = \log m + \sum_{i=1}^m p_i\log q_i. $$