Refined distributional approximations for the uncovered set in the Johnson–Mehl model

Let Φz be the uncovered set (i.e., the complement of the union of intervals) at time z in the one-dimensional Johnson–Mehl model. We derive a bound for the total variation distance between the distribution of the number of components of Φz∩(0,t] and a compound Poisson-geometric distribution, which is sharper and simpler than an earlier bound obtained by Erhardsson. We also derive a previously unavailable bound for the total variation distance between the distribution of the Lebesgue measure of Φz∩(0,t] and a compound Poisson-exponential distribution. Both bounds are O(zβ(t)/t) as t→∞, where zβ(t) is defined so that the expected number of components of Φzβ(t)∩(0,t] converges to β>0 as t→∞, and the parameters of the approximating distributions are explicitly calculated.