Limit theorems for continuous time random walks with slowly varying waiting times
Continuous time random walks are used in physics to model particle motion assuming random waiting times between the random jumps of a particle. The authors introduce the new feature of i.i.d. waiting times with a slowly varying probability tail. In this case the asymptotic waiting time process is known to be a Fréchet-type extremal process. In a corresponding limit theorem, the hitting time process is shown to have exponentially distributed marginals and increments with a defective exponential distribution. For an operator stable limit process of the i.i.d. jumps, assuming jumps and waiting times to be independent, an application of an appropriate transfer theorem proves a functional limit theorem for the continuous time random walk. Unlike the case of regular varying power law tails of the waiting times, the defectiveness of hitting time increments implies that particles may rest for a random time period in the resulting limiting process. The presented results are thus useful in physics to model ultraslow diffusions.
scientific article; zbMATH DE number 2142302