Stochastic discrete scale invariance and Lamperti transformation

Abstract

We define and study stochastic discrete scale invariance (DSI), a property which requires invariance by dilation for certain preferred scaling factors only. We prove that the Lamperti transformation, known to map self-similar processes to stationary processes, is an important tool to study these processes and gives a more general connection: in particular between DSI and cyclostationarity. Some general properties of DSI processes are given. Examples of random sequences with DSI are then constructed and illustrated. We address finally the problem of analysis of DSI processes, first using the inverse Lamperti transformation to analyse DSI processes by means of cyclostationary methods. Second we propose to re-write these tools directly in a Mellin formalism.

Stochastic discrete scale invariance

“…The next step is the investigation of the proposed framework for estimation on real data. First attempts, proposing definitions, and analysis of simple sequences with DSI were reported in [21]. These methods will now enable to reissue the relevance of DSI [3] (e.g., as found in geophysics where evidence of DSI in earthquakes was given, or in the DLA model for growth phenomena), by examining directly in the time domain the underlying DSI processes attached to the problems under consideration.…”