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

Let [Phi]z 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 [Phi]z[intersection](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 [Phi]z[intersection](0,t] and a compound Poisson-exponential distribution. Both bounds are O(z[beta](t)/t) as t-->[infinity], where z[beta](t) is defined so that the expected number of components of [Phi]z[beta](t)[intersection](0,t] converges to [beta]>0 as t-->[infinity], and the parameters of the approximating distributions are explicitly calculated.