  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 Cerny [Comm.
Math. Phys. 282 (2008) 663-695] that obtained similar convergence results in
law, with respect to the random environment.
