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Summary: IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005 1063
On Non-Scale-Invariant Infinitely Divisible Cascades
Pierre Chainais, Rudolf Riedi, and Patrice Abry
Abstract--Multiplicative processes, multifractals, and more
recently also infinitely divisible cascades have seen increased
popularity in a host of applications requiring versatile multiscale
models, ranging from hydrodynamic turbulence to computer
network traffic, from image processing to economics. The method-
ologies prevalent as of today rely to a large extent on iterative
schemes used to produce infinite detail and repetitive structure
across scales. While appealing, due to their simplicity, these
constructions have limited applicability as they lead by default
to power-law progression of moments through scales, to non-
stationary increments and often to inherent log-periodic scaling
which favors an exponential set of scales. This paper studies and
develops a wide class of infinitely divisible cascades (IDC), thereby
establishing the first reported cases of controllable scaling of
moments in non-power-law form. Embedded in the framework
of IDC, these processes exhibit stationary increments and scaling
over a continuous range of scales. Criteria for convergence, further
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