We introduce a family of random locations called "intrinsic location functionals". This family contains and extends far beyond many commonly used locations, such as the loca-tion of the path supremum in a compact interval, the first/last hitting time, etc. On one hand, we prove that under stationarity, the distributions of these random locations must satisfy a group of very specific conditions, including absolute continuity, certain constraints on the total variation of the density functions, among others. On the other hand, this group of conditions actually fully characterizes stationarity, in the sense that a stochastic process is stationary if and only if the conditions hold for all intrinsic location functionals. This provides a new approach to understand stationarity from the perspective of random locations. We further develop structural results for the set of all possible distributions for intrinsic location functionals under stationarity, and derived related optimal inequalities for the expectations of functions of random locations. This is a joint work with Gennady Samorodnitsky.
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