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 transformation to analyse DSI processes by means of cyclostationary methods. Second we propose to re-write these tools directly in a Mellin formalism. 1. DISCRETE SCALE INVARIANCE Scale invariance, also called self-similarity, is frequently called upon. Its central point is that the signal is scale invariant if it is equivalent to any of its rescaled versions, up to some amplitude renormalization [1]. More precisely, a function is scale-invariant with exponent , or -ss, if for any : This definition is given here for a deterministic signal. The concept can be extended to stochastic signals when one thinks of the previous equality in a probabilistic way: the equality of the finite-dimensional probability distributions [1]. We will write this equality. The strict notion of scale invariance, valid for all dilation factors above, is in some cases too rigid; the middlethird Cantor set is for example invariant only by dilations of a factor 3 (or a power of 3). Several weakened versions of self-similarity have been proposed to enlarge scale invariance’s relevance and one is of special interest here: it is to require invariance by dilation for certain preferred scaling factors only, as it is the case for the Cantor set. This is known as discrete scale invariance (DSI), a concept which as been stressed upon by Sornette and Saleur [2, 3] as an efficient model in many situations (fracture, DLA, critical phenomena, earthquakes). They studied DSI as a property of deterministic signals, and provided general arguments as why should DSI naturally occur: classical scenarii involve the existence of a characteristic scale, the apparition by instability of a preferred scale or more general arguments in non-unitary field theories [4]. They also found ways to estimate the preferred scaling ratio in this context, based on classical spectral analysis (Lomb periodogram). As far as we know, this property has not been envisioned for stochastic processes, a framework which is often fruitful to dispose of when dealing with real measurements, as it allows to use statistical signal processing methods. The extension of DSI property to stochastic processes is straigthforward. We propose the following definition. A process has discrete scale invariance with scaling exponent and scale if (1) We will refer to this property as -DSI. The equality here is the probabilistic equality. In the following only wide-sense property will be used (second-order statistical properties only). 2. LAMPERTI TRANSFORM : DSI AS AN IMAGE OF CYCLOSTATIONARITY 2.1. Lamperti transformation A main issue is to find a way to study both theoretically and practically DSI processes. The answer is given by a transformation introduced by J. Lamperti in 1962 [5], which is an isometry between self-similar and stationary processes. It will be called the Lamperti transformation and is defined as follows. For any process "! # $ , its Lamperti transform