Does local martingale have the same mean value as well? We know that if $\{M_n\}$ is a martingale, we know from definition of martingale that $E(M_n) = M_0$ for all $n \geq 0$. However, if we only know that a sequence of random variables $\{X_n\}$ is a local martingale, do we still have the result that
$$E(X_n) = X_0, \forall n\geq 0?$$
And why?

Based on the first paragraph of Wiki, no:

Every martingale is a local martingale; every bounded local martingale
  is a martingale; in particular, every local martingale that is bounded
  from below is a supermartingale, and every local martingale that is
  bounded from above is a submartingale; however, in general a local
  martingale is not a martingale, because its expectation can be
  distorted by large values of small probability. In particular, a
  driftless diffusion process is a local martingale, but not necessarily
  a martingale.

