Some results on joint record events

Abstract

Let X1,X2,… be independent and identically distributed random variables on the real line with a joint continuous distribution function F. The stochastic behavior of the sequence of subsequent records is well known. Alternatively to that, we investigate the stochastic behavior of arbitrary Xj,Xk,j<k, under the condition that they are records, without knowing their orders in the sequence of records. The results are completely different. In particular it turns out that the distribution of Xk, being a record, is not affected by the additional knowledge that Xj is a record as well. On the contrary, the distribution of Xj, being a record, is affected by the additional knowledge that Xk is a record as well. If F has a density, then the gain of this additional information, measured by the corresponding Kullback–Leibler distance, is j∕k, independent of F. We derive the limiting joint distribution of two records, which is not a bivariate extreme value distribution. We extend this result to the case of three records. In a special case we also derive the limiting joint distribution of increments among records.