Random multiplicative multifractal measures. II: Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures. (English) Zbl 1117.28007
Lapidus, Michel L. (ed.) et al., Fractal geometry and applications: A jubilee of Benoît Mandelbrot. Multifractals, probability and statistical mechanics, applications. In part the proceedings of a special session held during the annual meeting of the American Mathematical Society, San Diego, CA, USA, January 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3638-2/v.2; 0-8218-3292-1/set). Proceedings of Symposia in Pure Mathematics 72, Pt. 2, 17-52 (2004).
[For part I see ibid., 3–16 (2004; Zbl 1088.28004).]
This paper consists of six sections. In Sections 2 and 3 the authors investigate non-degeneracy and \(L^p\) convergence of a wide class of random multiplicative measures including the “Multifractal products of cylindrical pules” (MPCP), “Canonical cascade measures” (CCM), “Poisson canonical cascade measures” (PCCM) and log-infinitely divisible cascades. Section 4 summarizes results on simultaneous convergence of uncountable families of non-degenerate martingale limiting measures. These results are useful in studying the multifractal analysis of CCM or MPCP, as well as in random covering problems.
Section 5 studies the non-degeneracy, moments, carrying dimension and multifractality of a subclass of the random multiplicative measures considered in Sections 2 and 3, namely, those satisfying certain self-similarity property in distribution. The results in this section are applied in Section 6 to MPCP, PCCM and CCM. The results summarized in this paper complete Kahane’s general theory on \(T\)-martingales.
[For part III (by the first author) see ibid., 53–90 (2004; Zbl 1117.28006).]
For the entire collection see [Zbl 1055.37003].
This paper consists of six sections. In Sections 2 and 3 the authors investigate non-degeneracy and \(L^p\) convergence of a wide class of random multiplicative measures including the “Multifractal products of cylindrical pules” (MPCP), “Canonical cascade measures” (CCM), “Poisson canonical cascade measures” (PCCM) and log-infinitely divisible cascades. Section 4 summarizes results on simultaneous convergence of uncountable families of non-degenerate martingale limiting measures. These results are useful in studying the multifractal analysis of CCM or MPCP, as well as in random covering problems.
Section 5 studies the non-degeneracy, moments, carrying dimension and multifractality of a subclass of the random multiplicative measures considered in Sections 2 and 3, namely, those satisfying certain self-similarity property in distribution. The results in this section are applied in Section 6 to MPCP, PCCM and CCM. The results summarized in this paper complete Kahane’s general theory on \(T\)-martingales.
[For part III (by the first author) see ibid., 53–90 (2004; Zbl 1117.28006).]
For the entire collection see [Zbl 1055.37003].
Reviewer: Yimin Xiao (East Lansing)
MSC:
| 28A80 | Fractals |
| 60G18 | Self-similar stochastic processes |
| 60G44 | Martingales with continuous parameter |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60G57 | Random measures |
