Fair-coin doubling martingale
= Fair-coin doubling martingale
For fair <independent and identically distributed random variables> $\xi_j$ with the <Bernoulli distribution>, $M_n=2^n\mathbf1_{\{\xi_1=\cdots=\xi_n=1\}}$ is a nonnegative <martingale>. It has <almost sure convergence> to zero but $\mathbb E M_n=1$, so it lacks <convergence in L1>.