What methods are there for showing a time series is mean reverting?
Is there a hypothesis relating to the Ornstein-Uhlenbeck process for example?
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In econometrics all these tests are under the banner of 'Unit Root Tests'. There is a vast body of literature that deals with their formulation, treatment and pifalls.
In essence, and without using any equations: A process that is mean reverting to zero will have a tendency to decrease when it is above zero and a tendency to increase when it is below zero. This can be tested by regressing the observed process changes during each time interval on the process state at the beginnning of each time interval.