From Stationarity to Self-similarity, and Back: Variations on the Lamperti Transformation

Abstract

The Lamperti transformation defines a one-to-one correspondence between stationary processes on the real line and self-similar processes on the real half-line. Although dating back to 1962, this fundamental result has further received little attention until a recent past, and it is the purpose of this chapter to survey the Lamperti transformation and its (effective and/or potential) applications, with emphasis on variations which can be made on the initial formulation. After having recalled basics of the transform itself, some results from the literature will be reviewed, which can be broadly classified in two types. In a first category, classical concepts from stationary processes and linear filtering theory, such as linear time-invariant systems or ARMA modeling, can be given self-similar counterparts by a proper “lampertization” whereas, in a second category, problems such as spectral analysis or prediction of self-similar processes can be addressed with classical tools after stationarization by a converse “delampertization”. Variations and new results will then be discussed by investigating consequences of the Lamperti transformation when applied to weakened forms of stationarity, and hence of self-similarity. Different forms of locally stationary processes will be considered this way, as well as cyclostationary processes for which “lampertization” will be shown to offer a suitable framework for defining a stochastic extension to the notion of discrete scale invariance which has recently been put forward as a central concept in many critical systems. Issues concerning the practical analysis (and synthesis) of such processes will be examined, with a possible use of Mellin-based tools operating directly in the space of scaling data.