It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We show that this distribution is absolutely continuous in the interior of the interval and describe very specific conditions the density has to satisfy. We establish universal upper bounds on the density and demonstrate their optimality.
"Is the location of the supremum of a stationary process nearly uniformly distributed?." Ann. Probab. 41 (5) 3494 - 3517, September 2013. https://doi.org/10.1214/12-AOP787