Some Results on Joint Record Events

Abstract

Let $X_1,X_2,\dots$ 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 $X_j,X_k,j<k$, 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 $X_k$, being a record, is not affected by the additional knowledge that $X_j$ is a record as well. On the contrary, the distribution of $X_j$, being a record, is affected by the additional knowledge that $X_k$ 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.