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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.