Records in subsets of a random field

Consider an i.i.d. random field {Xk:k∈Z+d}, together with a sequence of unboundedly increasing nested sets Sj=⋃k=1jHk,j≥1, where the sets Hj are disjoint. The canonical example consists of the hyperbolas Hj={k∈Z+d:|k|=j}. We are interested in the number of “hyperbolas” Hj that contain at least one record, and, furthermore, the number of records on the “next hyperbola”, that is, the number of observations on Hj that exceed max{Xk:k∈Sj−1}. Various limit theorems under mild conditions on the size of the sets Hj are presented.