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1

I don't think you can say anything general on this type of setup, certainly not from an empirical point of view. Assume the market conditions change between the two periods, then $ES$ could be higher or lower. If you assume some distribution for you returns, then they should probably be the same if the two periods have the same length.

2

This reminds me of a paper by Rama Cont: "Empirical properties of asset returns: stylized facts and statistical issues.". You can download here: http://www.cmap.polytechnique.fr/~rama/papers/empirical.pdf He also has a paper on volatility clustering: "Volatility clustering in financial markets: empirical facts and agent-based models.", which may be of your ...

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Stationarity. The distribution of returns is non-stationary. Moreover, standard deviation of returns is not constant over time. Symmetry. The distribution of returns is approximately symmetric with increasing leptokurtosis as sampling frequency increases. However, large drawdowns are not matched with equally large upward movements. Gaussian behavior. ...

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You may want to take a look at lagged correlation or cross correlation. Lagged correlation refers to the correlation between two time series shifted in time relative to one another. This measure is useful for studying whether a lagged time series $x_{t-k}$ can be viewed as a good predictor for $y_t$. If you are familiar with R, then you may find the ...

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