How to detect if Ergodicity, Stationarity and Martingale. dif. sequence are satisfied? I'm not sure, but I think I've read somewhere that because the Classical Linear Regression model assumes to have a random sample, when researchers think they might not be in presence of a sample with that property, they try to use some randomization technique to make sure the usual theory may be applied.

In Hayashi's Econometrics, chapter 2, he develops the OLS and studies its large-sample properties in a generalization of iid sample. He assumes that the sample is ergodic, and stationary, and that the regressors multiplied by the error terms (the same that define the orthogonality conditions) also follow a martingale difference sequence.

My question is, given a sample, is there a way to know if it satisfies ergodicity, stationarity, and martingale diff. seq.? Also, even if the sample does not satisfy it, is there a way to make sure that we are able to obtain such a sample, when randomization techniques are not possible to apply?

Any help would be appreciated.

P.S.:
This question is also posted in CrossValidated, but with a different intention. When posting it here, I'm looking more for a perspective of applied econometricians, not so much for formal mathematical explanation of the possible existing methods. But of course, if you're willing to formalize and going into depth I would also be very thankful.
Ergodic and strict stationarity are the essentially the weakest assumptions for which you have a LLN, i.e. can do large sample estimation. Given the very liberal way applied econometricians use laws of large numbers, ergodicity and strict stationarity is almost always assumed. If that is in question, you're simply not in business for consistent estimation.

Similarly, if one would like to have a CLT, i.e. do large sample inference, MDS is a general weak assumption. (There are other CLT's around but more technical. The MDS CLT and Lindenberg CLT are the two immediate generalizations of the classical CLT.)

The MDS assumption can be tested. An MDS is just a discrete-time martingale with mean zero. In particular, its increments are uncorrelated. (The martingale assumption is an intermediate condition between serially uncorrelated and independence.) So a rejection by a serial correlation test would also reject the MDS assumption.