Hi guys,
I am trying to prove the L2 Martingale Convergence Theorem but am struggling with some real analysis.
For a martingale {M_n} such that E[|M_n|^2] \le B < \infty,
let
A_ab = {\omega: lim inf M_n(\omega) \le a < b \le lim sup M_n(\omega)}
How is
A_ab \in {\omega: sup_{m\le k < \infty} |M_k - M_m| \ge \epsilon}
for \epsilon > (b-a)/2?
Thank you for any advice in advance.
Regards,
MancBluebird