On multivariate records from random vectors with independent components

While there are infinitely many records in univariate case, there occur only finitely many in the case of multivariate series. Consequently, there is a terminal complete record with probability one. We compute the distribution of the random total number of complete records and investigate the distribution of the terminal record. For complete records, the sequence of waiting times forms a Markov chain, but differently from the univariate case, now the state infinity is an absorbing element of the state space.