Convergence of a typical martingale (A remark on the Doob theorem)

Abstract

We study convergence behavior of discrete martingales with values in the interval [0,1] from a measure theoretical point of view as well as from a topological one. We show that almost all martingales converge to 0 or 1 almost everywhere.

On the other hand, a typical martingale diverges on a comeager set.

Keywords

• Martingale
• Typical behavior
• Doob theorem