$\zeta$-functions and the topology of superlevel sets of stochastic processes

Abstract

We describe the topology of superlevel sets of ($\alpha$-stable) L\'evy processes X by introducing so-called stochastic $\zeta$-functions, which are defined in terms of the widely used $\text{Pers}_p$-functional in the theory of persistence modules. The latter share many of the properties commonly attributed to $\zeta$-functions in analytic number theory, among others, we show that for $\alpha$-stable processes, these (tail) $\zeta$-functions always admit a meromorphic extension to the entire complex plane with a single pole at $\alpha$, of known residue and that the analytic properties of these $\zeta$-functions are related to the asymptotic expansion of a dual variable, which counts the number of variations of X of size $\geq \varepsilon$. Using these results, we devise a new statistical parameter test using the topology of these superlevel sets. We further develop an analogous theory, whereby we consider the dual variable to be the number of points in the persistence diagram inside the rectangle $]\infty, x]\times[x+\varepsilon, \infty[$.