The Almost Lack of Memory (ALM) distributions and their applications

Abstract

This article generalizes fundamental research devoted to a class of probability distributions known as Almost Lack of Memory (ALM) distributions. While the classical exponential distribution is uniquely characterized by the lack of memory property at any age, ALM distributions exhibit this property selectively, at ages forming a discrete sequence {0,  c, 2c, ...} for a positive constant c. This generalization provides a powerful and versatile tool for modeling lifetimes, times to failure, and waiting times under periodic, non-stationary conditions. We present a comprehensive characterization of ALM distributions through several equivalent properties: a specific analytical form of the distribution function, a periodic failure rate, and a probabilistic representation in which a random variable decomposes into the sum of a component truncated on [0, c) and a geometrically distributed component scaled by c. This decomposition is of key importance for both theoretical analysis and practical applications. The intrinsic connection between ALM distributions and periodic non-homogeneous Poisson processes is rigorously established. Furthermore, the article discusses important statistical aspects, including parameter estimation by maximum likelihood and method of moments, as well as hypothesis testing for model discrimination. The applicability of ALM theory is demonstrated in various fields, including the reliability of technical systems operating under cyclic loads, insurance for modeling losses with a seasonal structure, and environmental sciences for studying phenomena driven by periodic forces. Bridging the gap between theoretical probability and the need for modeling real-world periodicity, ALM distributions offer a robust and mathematically elegant tool for analysis under non-stationary conditions.