On Power Series Related to Multivariate Records

The paper introduces a function defined by a power series in the probabilities of multivariate records in a sequence of independent identically distributed random vectors with continuous components. In the univariate case, this function is always the same, but in the multivariate case there exists a broad variety of such functions determined by distribution copulas. The probabilistic meaning of this function and its derivatives is presented. A calculation method using the Kendall distribution function is given. The concepts of favorableness of copulas for records, record time distribution, and mean record time (without taking into account the order) are introduced. Examples are given.