# Is Martingale difference sequence strictly stationary and ergodic?

1. Mar 26, 2008

### CHatUPenn

Is Martingale difference sequence strictly stationary and ergodic?
It seems to me that Martingale Difference Sequence is a special case of strictly stationary and ergodic sequences.

Also, can somebody give me an example of strict stationarity without independence.

Cheers

2. Apr 13, 2008

### sandhammaren

Martingales are nonstationary

Martingales are nonstationary processes (with one singular exception) and martingale differences/increments are generally nonstationary. If the increments are stationary then there is (i) no ergodicity and (ii) no i.i.d. unless the diffusion coefficient is both time and space translationally invariant. When that holds, one has the wiener process and the Markov condition on the transition density yields i.i.d., which yields ergodicity (convergence of time averages of increments to ensemble average of zero). For a general stationary increment martingale process, time averages do not converge.

Example of a non i.i.d. stationary process: (i) the Ornstein-Uhlenbeck process, and (ii) in discrete time y(t)=ay(y-T) +e(t), T fixed and e(t) is uncorrelated, if 0<a<1.

Reference: recent papers by McCauley, Gunaratne, and Bassler