Generalized Lamperti transformation of broken scale invariance

The Lamperti transformation is a powerful tool for studying nonstationary self-similar processes via associated stationary generators. After having discussed how the Lamperti transformation offers a new perspective on the representation of nonstationary self-similar processes, we propose here to enlarge its scope in two manners. First, we show that many kinds of broken self-similarity admit Lamperti counterparts which are weakened forms of stationarity: a prominent example is that of discrete scale invariance (i.e., scale invariance for some preferred scaling factors only), which is the Lamperti image of cyclostationarity. Second, we introduce modified versions of the original Lamperti transformation in order to stationarize other forms of broken scale invariance such as local self-similarity or finite-size scale invariance based on warped dilations properties.