A Player starts with start capital $C_0=1$. Each round $i=1,...,n$ he bets half of his capital. Every round there is a fair coin toss. For "Head" he will get double the amount he did bet, for "Tail" he will lose the capital he did bet.
- Now what I find difficult is first to model this the right way. How do I have to choose $(\Omega,F,P)$ and why? This is an infinite game, that's difficult for me.
- How high is the amount of Money the Player has after $n$ rounds, that might be $K_n=\prod_{i=1}^n R_i, n\in \mathbb{N}$
- I know that $\mathbb{E}[K_n]=1$ but how do I prove that? I guess I need to use the law of large numbers here right?
- Last Question: Why is $\lim_{n \rightarrow\infty}K_n=0$ almost surely?
For the second bullet, you have not defined $R_i$. Note that if he wins his capital is multiplied by $\frac 32$ while if he loses his capital is multiplied by $\frac 12$. After $n$ rounds if he has won $k$ times his capital is multiplied by $\left(\frac 32\right)^k\left(\frac 12\right)^{n-k}$
For the third bullet, each bet is fair, so the expected value cannot change.
For the fourth bullet, in the long run $k \approx n-k$ and the power of $\frac 34$ will dominate. The expectation remains $1$ because there is a tiny chance he has a huge amount of money.