Some Results on Joint Record Events

TL;DR

This paper investigates the stochastic behavior of arbitrary record pairs in i.i.d. sequences, revealing that knowledge of one record's distribution does not affect the other, and derives their joint distributions and information gains.

Contribution

It provides new insights into the joint distribution of arbitrary record pairs, contrasting with the well-known behavior of sequential records, and extends results to multiple records and increments.

Findings

Distribution of a record is unaffected by knowledge of another record's occurrence.

Distribution of the other record is influenced by the knowledge of the first.

Additional information gain measured by Kullback-Leibler distance is j/k, independent of the distribution.

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.