Stochastic discrete scale invariance and Lamperti transformation

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( 1962) transformation to analyse DSI processes by means of cyclostationary methods. Second we propose to re-write these tools directly in a Mellin formalism.