Many multivariate records

(Xn)n=1,2,… is a sequence of independent, identically distributed random vectors in R2. Say Xn is a record if there is a record simultaneously in both coordinates at index n. The total number of records of a sequence is frequently finite so we consider the behaviour of the records in a fixed rectangle A conditional on there being a large number of records NA in the rectangle. We join the records up to form a random parametrized curve, and prove that, under suitable conditions, as the number of records in the rectangle goes to infinity the random curves converge conditionally in probability to a non-random limit parametrized curve, which we characterize. We also prove a large-deviation result for the random curves.