On the relationships between the extreme values of two-dimensional random vectors (Q1086907)

The extreme values of a sequence of two-dimensional random vectors are studied. The elements of the sequence are supposed to be associated with the ''births'' of a homogeneous Poisson process. If X and Y are the (independent) components of the random vectors and if N(t) is the amount of the population at the time t, then it is shown that while the conditional record values \(\{M_ x(t)| N(t)=n\}\) and \(\{M_ y(t)| N(t)=n\}\) are independent for every \(n\geq 0\), the unconditional record values \(M_ x(t)\) and \(M_ y(t)\) are not. The probability that one record is bigger than the other is given in section 3. It is also pointed out that if at the time t we have \(M_ x(t)<M_ y(t)=v\), then, no matter how big v is, there is always a positive probability that sooner or later the inequality between the two records will be reversed. In section 5 the problem of which one of the two records will be more likely the first to be broken is tackled. In section 4 the waiting time for the record \(M_ x(s)\) \((M_ y(s))\) to be broken is introduced. It is demonstrated that the distribution function of such a waiting time does not depend on the distribution function of X (Y) so that the waiting time for \(M_ x(s)\) to be broken has the same distribution function as the waiting time for \(M_ y(s)\) to be broken. By means of this result it is also proved that two counting processes associated with the extreme values of X and Y have a common counting measure.