Records in stochastic processes-theory and applications (Q2837781)

This paper reviews mathematical results on records in five sections with different topics. Section 2 provides an introduction to the classical theory of records within a sequence \(X_0,X_1,\dots\) of independent and identically distributed random variables with a continuous distribution function, where \(X_n\) is a record if \(X_n>\max(X_0,X_1,\dots,X_{n-1})\), \(n\in \mathbb{N}\). Section 3 describes recent results on records for independent but non-identically distributed \(X_0,X_1,\dots\), in particular where the random variables follow a linear drift model or an increasing variance model. Section 4 is on records from a discrete distribution, rounded records, \(\delta\)-records and geometric records. Section 5 deals with records from dependent observations, in particular from discrete and continuous random walks. Applications of the theory of records to climate and financial data, physics, biology and sports are listed in Section 6. Open problems are formulated in the final Section 7.