On functional records and champions

TL;DR

This paper extends the classical concept of records from sequences of iid random variables to multivariate and stochastic process settings, introducing simple and complete records and analyzing their asymptotic behavior.

Contribution

It proposes two new definitions of records in higher dimensions and stochastic processes, and studies their probabilities and distributions under max-stable process assumptions.

Findings

Probability of records as n→∞ under max-domain attraction

Distribution of a record given it occurs

Differences between simple and complete records

Abstract

Records among a sequence of iid random variables $X_1,X_2,\dotsc$ on the real line have been investigated extensively over the past decades. A record is defined as a random variable $X_n$ such that $X_n>\max(X_1,\dotsc,X_{n-1})$. Trying to generalize this concept to the case of random vectors, or even stochastic processes with continuous sample paths, the question arises how to define records in higher dimensions. We introduce two different concepts: A simple record is meant to be a stochastic process (or a random vector) $X_n$ that is larger than $ X_1,\dotsc, X_{n-1}$ in at least one component, whereas a complete record has to be larger than its predecessors in all components. The behavior of records is investigated. In particular, the probability that a stochastic process $ X_n$ is a record as $n$ tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the distribution of $ X_n$, given that $ X_n$ is a record is derived.