Correlations between record events in sequences of random variables with a linear trend

The statistics of records in sequences of independent, identically distributed random variables is a classic subject of study. One of the earliest results concerns the stochastic independence of record events. Recently, records statistics beyond the case of i.i.d. random variables have received much attention, but the question of independence of record events has not been addressed systematically. In this paper, we study this question in detail for the case of independent, non-identically distributed random variables, specifically, for random variables with a linearly moving mean. We find a rich pattern of positive and negative correlations, and show how their asymptotics is determined by the universality classes of extreme value statistics.

💡 Research Summary

The paper investigates how record events—instances where a new observation exceeds all previous ones—are correlated when the underlying random variables are independent but not identically distributed, specifically when each variable carries a linear drift in its mean. In the classic i.i.d. setting, record events are known to be stochastically independent; the authors revisit this result and then introduce the Linear Drift Model (LDM) defined by Yₗ = Xₗ + cℓ, where the Xₗ are i.i.d. with a common density f(x) and c>0 is a constant drift per time step.

The central quantity studied is the normalized joint probability

 lₙ,ₙ₋₁(c) = pₙ,ₙ₋₁(c) /