Abstract
A notion of semi-selfsimilarity of R d-valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.
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Maejima, M., Sato, Ki. Semi-Selfsimilar Processes. Journal of Theoretical Probability 12, 347–373 (1999). https://doi.org/10.1023/A:1021621926463
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DOI: https://doi.org/10.1023/A:1021621926463