Testing for stationarity in large sample sizes All you need is to spend more time getting familiar with the definition of stationary process with some examples and counterexamples. It is standard material in any time series analysis book.
How do you estimate the capacity of a strategy from historical data? +1, good question. Can you expand on why standard market impact modeling (such as that described here) is not adequate? It seems to me that the modeling tools would be the same, and what would change is instead the maximization problem implied by the strategy (in contrast with the simpler "execute a large block").
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