Self-Similarity and Lamperti Convergence for Families of Stochastic Processes

TL;DR

This paper introduces a new form of self-similarity for families of stochastic processes, including fractional Hougaard motions and Hougaard Lévy processes, with implications for their covariance structures and limit theorems.

Contribution

It defines a novel type of self-similarity applicable to various important stochastic process families, extending classical concepts.

Findings

New class of fractional Hougaard motions as moving averages of Hougaard Lévy processes

Identification of common properties with self-similar processes, including covariance forms

Establishment of Lamperti-type limit theorem for these process families

Abstract

We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to a number of important families of processes that are not self-similar in the conventional sense. This includes a new class of fractional Hougaard motions defined as moving averages of Hougaard L\'evy process, as well as some well-known families of Hougaard L\'evy processes such as the Poisson processes, Brownian motions with drift, and the inverse Gaussian processes. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.