Records for Some Stationary Dependent Sequences

TL;DR

This paper derives probabilities and distributions related to records in stationary Gaussian processes, including joint distributions, expected number of records, and asymptotic behaviors, extending results to multivariate cases and processes with long-range dependence.

Contribution

It provides new explicit formulas and asymptotic results for record probabilities in stationary Gaussian and multivariate Gaussian processes, including dependence and tail behavior.

Findings

Derived probability and distribution of record occurrences in Gaussian processes

Extended results to multivariate Gaussian processes

Established asymptotic behavior for records in long-range dependent processes

Abstract

For a zero-mean, unit-variance second-order stationary univariate Gaussian process we derive the probability that a record at the time $n$, say $X_n$, takes place and derive its distribution function. We study the joint distribution of the arrival time process of records and the distribution of the increments between the first and second record, and the third and second record and we compute the expected number of records. We also consider two consecutive and non-consecutive records, one at time $j$ and one at time $n$ and we derive the probability that the joint records $(X_j,X_n)$ occur as well as their distribution function. The probability that the records $X_n$ and $(X_j,X_n)$ take place and the arrival time of the $n$-th record, are independent of the marginal distribution function, provided that it is continuous. These results actually hold for a second-order stationary process with Gaussian copulas. We extend some of these results to the case of a multivariate Gaussian process. Finally, for a strictly stationary process satisfying some mild conditions on the tail behavior of the common marginal distribution function $F$ and the long-range dependence of the extremes of the process, we derive the asymptotic probability that the record $X_n$ occurs and derive its distribution function.