bootstrap and non-stationarity

[resetzero]

Dear all,

Is it true to say:

If a time series X(t) does not satisfy the strict stationarity condition:

P(X(t+1), X(t+2), ... , X(t+T)) =/= P(X(t+1+k), X(t+2+k), ... , X(t+T+k))

where t and k are time and time difference/shift respectively and =/= is not equal to, then NO bootstrap algorithm (block or otherwise) can resample the series such that the strict stationarity condition is satisfied:

P(X*(t+1), X*(t+2), ... , X*(t+T)) = P(X*(t+1+k), X*(t+2+k), ... , X*(t+T+k))

where X* is the bootstrapped observation. That is, the strict stationarity condition implies that the joint distribution of the series only depends on the time difference, k, and not on time, t.

For instance, one can imagine X to be generated by a randomly changing data generation process.

Thanks,
RZ