Characterisation of distributions through $\delta$-records and martingales

M. Lafuente Blasco, R. Gouet Bañares, F. J. López Lorente, G. Sanz Sáiz

Our aim is to characterise distributions $F$ such that the process $(N_n-c M_n)$ is a martingale, where $N_n$ counts the number of $\delta$-records —observations exceeding the previous maximum by at least $\delta$—and $M_n$ is the running maximum of an i.i.d. sequence $(X_n)$. Given parameters $c>0$, $\delta \neq 0$, this problem is recast as $1-F(x+\delta) = c \int_{x}^{\infty} (1-F)(t) dt, \ x \in T,\ F(T) = 1.$

Unlike standard functional equations, the domain $T$ depends on $F$, which introduces complexity but allows for a wider range of solutions.

We find the explicit expressions of all solutions when $\delta<0$ and, when $\delta>0$, for distributions with bounded support. In the unbounded case, the problem reduces to a delay differential equation for continuous distributions on $\mathbb{R}_+$ and a difference equation for lattice distributions on $\mathbb{Z}_+$. Also, we provide explicit examples and establish necessary and sufficient conditions for the existence of solutions.

Keywords: Characterisation of distributions, Martingales, Records, δ-records, Delay differential equation

Stochastic processes and their applications IV
June 13, 2025  9:00 AM
Auditorio 2. Leandre Cristòfol

M. González Velasco, I. del Puerto García

J. A. Vega Coso, J. Villarroel

M. Gamboa Pérez, M. J. López Herrero

J. Villarroel, D. Hernández-Serrano