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I have a time series $(x_i)_{i=0}^n$ and want to know if a positive (resp. negative) increment is more likely to be followed by another positive (resp. negative) increment or not.

The naïve statistic to use is $$S=\frac{\sum_{i=1}^{n-1}\mathrm{sign}(x_{i+1}-x_i)\cdot\mathrm{sign}(x_i-x_{i-1})}{n-1}.$$

Is this a known statistic? Is its distribution known?

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