  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.
