The probability of eventual return is given by:

$\displaystyle f_{2} + f_{4} + f_{6} + ...+ f_{\infty} = 1$

where $\displaystyle f_{n}$ = Probability of return at Time Period n.

Note that returns must be in even numbers, as a +1 needs -1 to cancel each other out.

Ok, I understand the theory, but I cannot find a suitable proof.

Edit: Proof found. Problem solved.

$\displaystyle f_{n}$ is given by $\displaystyle f_{2n}= u_{2n-2} - u_{2n}$ where $\displaystyle u_{2n}={2n \choose n}2^{-2n}$

So, $\displaystyle f_{2} + f_{4} + f_{6} + ...+ f_{\infty} = u_{0} - u_{2} + u_{2} - u_{4} + ..... = u_{0} = 1$

Still, I'll be very grateful if anyone can explain this Theorem:

$\displaystyle f_{2n}= u_{2n-2} - u_{2n}$