Limit laws for partial match queries in quadtrees

It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a first-order asymptotic expansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.