View Submission - CMStatistics

B1184
**Title: **Some results on joint record events
**Authors: **Amir Khorrami Chokami - University of Turin (Italy) **[presenting]**

Michael Falk - University of Wuerzburg (Germany)

Simone Padoan - Bocconi University (Italy)

**Abstract: **Let $\boldsymbol{X}_1,\boldsymbol{X}_2,\dots$ be i.i.d. copies of a r.v. $\boldsymbol{X}\in\mathbb{R}^d$ with a continuous joint distribution function $F$. When $d=1$, the stochastic behavior of the sequence of subsequent records is well known. Differently, we study the stochastic behavior of arbitrary $X_j,X_k,j<k$, under the condition that they are records, without knowing their order 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. The distribution of $X_j$, being a record, is however 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 independent of $F$.In the multivariate case, a random vector is defined a complete record if each component is a univariate record. In the case of independent components, it is known that there are finitely many complete records. New results on the arrival times of complete records are provided, and it is investigated the distribution of the terminal record.

Michael Falk - University of Wuerzburg (Germany)

Simone Padoan - Bocconi University (Italy)