The approximations of risk processes with mixed exponentially distributed inter-arrival times are investigated. The number of claims in a fixed time interval is mixed Poisson distributed. The approximating process is always overdispersed. This allows a better fit to more realistic situations in finances, than e.g. classical Cramer-Lundberg model. The claim sizes are divided in three different groups, dependently on finiteness of their first two moments. We illustrate all the cases by numerical examples. The case of diffusion approximation is investigated. Both American and European Pareto Claims sizes are also studied.