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Let $\boldsymbol{X}_1,\boldsymbol{X}_2,\dots$ be independent copies of arandom vector $\boldsymbol{X}$ with values in $\mathbb{R}^d$ and with acontinuous distribution function. The random vector $\boldsymbol{X}_n$ is acomplete record, if each of its components is a record. As we require$\boldsymbol{X}$ to have independent components, crucial results for univariaterecords clearly carry over. But there are substantial differences as well:While there are infinitely many records in case $d=1$, there occur onlyfinitely many in the series if $d\geq 2$. Consequently, there is a terminalcomplete record with probability one. We compute the distribution of the randomtotal number of complete records and investigate the distribution of theterminal record. For complete records, the sequence of waiting times forms aMarkov chain, but differently from the univariate case, now the state infinityis an absorbing element of the state space.

On Multivariate Records from Random Vectors with Independent Components