Fractional Brownian motion with Hurst exponent H has persistence probability — the probability of staying positive up to time T — that decays as T^(-(1-H)). Higher H means stronger positive correlations, longer excursions above zero, slower decay. The exponent is known exactly for fixed H (Molchan, 1999).
The paper considers the case where H itself is random. Instead of a fixed Hurst exponent, each realization of the process draws H from a distribution and then evolves as fractional Brownian motion with that H. The persistence probability of this mixture decays as T^(-(1-Hâ‚€)), where Hâ‚€ is the essential supremum of the distribution of H — the largest value that H can take with nonzero probability.
The result is clean: the persistence is dominated by the most persistent component. Among all the possible Hurst exponents, the one that produces the slowest decay — the largest H — determines the mixture's long-time behavior. The other components decay faster and become negligible. In the long-time limit, the mixture behaves as if only the most persistent component existed.
This is an extreme-value phenomenon in parameter space rather than sample space. The usual extreme-value theory asks which realizations of a fixed process dominate at long times. Here the question is which values of a random parameter dominate. The answer is the same: the most extreme parameter wins, because its contribution decays slowest and eventually drowns out everything else. Disorder in parameters produces the same winner-take-all structure as disorder in outcomes.