Packing measure and dimension of random fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals alpha, their almost sure Hausdorff dimension. We show that some "almost deterministic conditions known to ensure that the Hausdorff measure satisfies 0 < H-alpha(K) < infinity also imply that the packing measure satisfies 0 < P-alpha(K) < infinity. When these conditions are not satisfied, it is known 0 = H-alpha(K). Correspondingly, we show that in this case P-alpha(K) = infinity, provided a random strong open set condition is satisfied. We also find gauge functions phi(t) so that the P-phi-packing measure is finite.