patter2809
New member
- Joined
- Mar 29, 2013
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- 17
Are all Martingale Difference Sequences ergodic?
By definition if {Xt} is an MDS, if E[Xt|Ft-1] = 0 for all t.
So, Cov(Xt,Xs) := E[XtXs] - E[Xt]E[Xs]
and wlog assume t > s, so,
E[XtXs] = E[E[XtXs|Ft-1]] = E[Xs E[Xt|Ft-1]] = E[Xs . 0] = 0
and E[Xt]E[Xs] = E[E[Xt|Ft-1]]E[Xs] = E[0]E[Xs] = 0, so surely Cov[Xt,Xs] = 0 for all t =/= s, and so surely, (autocorrelation function) γ(t-s) => 0 as (t-s) => infinity, which implies ergodicity.
Thanks in advance!
By definition if {Xt} is an MDS, if E[Xt|Ft-1] = 0 for all t.
So, Cov(Xt,Xs) := E[XtXs] - E[Xt]E[Xs]
and wlog assume t > s, so,
E[XtXs] = E[E[XtXs|Ft-1]] = E[Xs E[Xt|Ft-1]] = E[Xs . 0] = 0
and E[Xt]E[Xs] = E[E[Xt|Ft-1]]E[Xs] = E[0]E[Xs] = 0, so surely Cov[Xt,Xs] = 0 for all t =/= s, and so surely, (autocorrelation function) γ(t-s) => 0 as (t-s) => infinity, which implies ergodicity.
Thanks in advance!