We construct random point processes in $\C$ that are asymptotically close to a given doubling measure. The processes we construct are the zero sets of random entire functions that are constructed through generalised Fock spaces. We offer two alternative constructions, one via bases for these spaces and another via frames, and we show that for both constructions the average distribution of the zero set is close to the given doubling measure. We prove some asymptotic large deviation estimates for these processes, which in particular allow us to estimate the `hole probability', the probability that there are no zeroes in a given open bounded subset of the plane. We also show that the `smooth linear statistics' are asymptotically normal, under an additional regularity hypothesis on the measure. These generalise previous results by Sodin and Tsirelson for the Lebesgue measure.