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I was working on a gambling problem where the gambler with an initial fortune $F_0$ stakes a proportion of $\theta$ of his current fortune (Billingsley Ex. 7.5). The wager $W_n=\theta F_{n-1}$ and they ask to show $F_n=\prod_{k=1}^{n}(1+\theta X_k)$ and $log F_n = n/2 \times [S_n/n \times log\{(1+\theta)/(1-\theta)\}+log(1-\theta^2)]$. Here, $S_n=X_1+\ldots+X_n$.

I can prove $F_n=\prod_{k=1}^{n}(1+\theta X_k)$ by induction. But I cannot see how taking the log of this gives me the expression mentioned above. I have also tried it from backward without any luck. Any thoughts?

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