We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [Ann. Probab. 6 (1978) 829–846]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the $p$-spin SK model. We prove that on a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent results of Ben Arous, Bovier and Černý [Comm. Math. Phys. 282 (2008) 663–695] that obtained similar convergence results in law, with respect to the random environment.
"Convergence of clock processes in random environments and ageing in the $p$-spin SK model." Ann. Probab. 41 (2) 817 - 847, March 2013. https://doi.org/10.1214/11-AOP705