Self-Similarity and Lamperti Convergence for Families of Stochastic Processes

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.