The location of the unique supremum of a stationary process on an interval does not need to be uniformly distributed over that interval. We describe all possible distributions of the supremum location for a broad class of such stationary processes. We show that, in the strongly mixing case, this distribution does tend to the uniform in a certain sense as the length of the interval increases to infinity.
"Distribution of the supremum location of stationary processes." Electron. J. Probab. 17 1 - 17, 2012. https://doi.org/10.1214/EJP.v17-2069