Characterisation of distributions via record-like observations
Raúl Gouet, Miguel Lafuente, F. Javier López, Gerardo Sanz
TL;DR
The paper studies when the number of δ-records $N_n$ adjusted by a linear term in the running maximum $M_n$ forms a martingale for i.i.d. observations with common distribution F. It reformulates the problem as a delay-integrated Cauchy functional equation $1 - F(x+δ) = c ∫_x^∞ (1-F(t)) dt$ on a distribution-dependent set, and analyzes continuous and discrete cases via delay differential and difference equations. A complete solution is obtained for δ<0 (all solutions are discrete with a specific structural form), while for δ>0 the authors identify all bounded-support solutions and provide a rich description for unbounded support through continuous DDEs and discrete recurrences, including mixtures of exponentials, gammas, and geometric/negative-binomial components. The results yield new characterisations, notably a novel geometric distribution characterisation based on weak records, and illuminate deep connections between martingale properties of δ-record counts and functional equations. The approach has implications for understanding limit behaviour of δ-record statistics and links to classical distributional characterisations.
Abstract
We characterise probability distributions via a martingale property associated with a natural generalisation of record values, known as $δ$-records. For an independent and identically distributed sequence $(X_n)$ with running maximum $M_n$, let $N_n$ be the number of $δ$-records (those $X_k$ with $X_k>M_{k-1}+δ$). We determine distributions for which $N_n-cM_n$ is a martingale, and show that this property uniquely determines the underlying distribution within broad classes. We show that the problem can be reformulated in terms of a delay-integrated Cauchy functional equation. A distinctive feature of this equation is that it is required to hold on a set that depends on the unknown distribution itself, which both complicates the analysis and allows for a rich variety of solutions. A complete characterisation is obtained when $δ<0$. For $δ>0$, all solutions with bounded support are identified. In the case of $δ>0$ and unbounded support, we consider both continuous and lattice distributions. In the continuous case, the characterisation reduces to a delay differential equation, which admits classical exponential-type solutions as well as broader families, including mixtures of exponential and gamma distributions. An analogous discrete analysis leads to difference equations whose solutions include mixtures of geometric and negative binomial distributions. In particular, this yields a new characterisation of the geometric distribution based on weak records.