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We derive a general criterion for the convergence of clock processes inrandom dynamics in random environments that is applicable in cases whencorrelations are not negligible, extending recent results by Gayrard [(2010),(2011), forthcoming], based on general criterion for convergence of sums ofdependent 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 caseof random hopping time dynamics of the p-spin SK model. We prove that on a widerange of time scales, the clock process converges to a stable subordinatoralmost surely with respect to the environment. We also show that a time-timecorrelation function converges to the arcsine law for this subordinator, almostsurely. This improves recent results of Ben Arous, Bovier and Cerny [Comm.Math. Phys. 282 (2008) 663-695] that obtained similar convergence results inlaw, with respect to the random environment.

Convergence of clock processes in random environments and ageing in the p-spin SK model