Densities of some Poisson $\mathrm{T}$-martingales and random covering numbers

Barral, Julien; Fan, Aihua

Comptes Rendus. Mathématique, Tome 338 (2004) no. 7, pp. 571-574.

Le comportement asymptotique du logarithme de la densité de certaines $\mathrm{T}$-martingales (au sens de la théorie de Kahane (Chinese Ann. Math. Ser. B 8 (1) (1987) 1–12)) est décrit de façon précise même en l'absence d'auto-similarité en loi. La construction de ces martingales fait intervenir des intensités de Poisson de la forme Lebesgue⊗μ sur $ℝ\times {ℝ}_{+}^{*}$. Nous montrons qu'il y a trois comportements possibles selon que $\overline{\alpha }=\lim {\sup }_{ϵ\to 0}{(-\log ϵ)}^{-1}{\int }_{[ϵ,1)}\ell \mathrm{d}\mu \left(\ell \right)$ est nul, strictement positif et fini ou infini. Cette question est intimement liée au comportements asymptotiques des nombres de recouvrements dans le recouvrement de Poisson pour la droite et le recouvrement de Dvoretzky pour le cercle.

The asymptotic behavior of the logarithm of the density of some $\mathrm{T}$-martingales (in the sense of Kahane theory (Chinese Ann. Math. Ser. B 8 (1) (1987) 1–12)) is described in detail even in absence of statistical self-similarity. Poisson intensities of the form Lebesgue⊗μ on $ℝ\times {ℝ}_{+}^{*}$ are involved in the construction of these martingales. We prove that there are three possible behaviors according to the fact that $\overline{\alpha }=\lim {\sup }_{ϵ\to 0}{(-\log ϵ)}^{-1}{\int }_{[ϵ,1)}\ell \mathrm{d}\mu \left(\ell \right)$ is zero, positive and finite, or infinite. This problem is closely related to the asymptotic behaviors of covering numbers in Poisson covering of the line and Dvoretzky covering of the circle.

[1] Bacry, E.; Muzy, J.-F. Log-infinitely divisible multifractal processes, Commun. Math. Phys., Volume 236 (2003), pp. 449-475

[2] Barral, J. Continuity of the multifractal spectrum of a statistically self-similar measure, J. Theoret. Probab., Volume 13 (2000), pp. 1027-1060

[3] Barral, J. Poissonian products of random weights: uniform convergence and related measures, Rev. Mat. Iberoamericana, Volume 19 (2003), pp. 813-856

[4] J. Barral, A.H. Fan, Covering numbers of different points in Dvoretzky covering, submitted for publication

[5] Barral, J.; Mandelbrot, B.B. Multifractal products of cylindrical pulses, Probab. Theory Related Fields, Volume 124 (2002), pp. 409-430

[6] J. Barral, B.B. Mandelbrot, Random multiplicative multifractal measures, Parts I, II, III, in: M.L. Lapidus, M. van Frankenhuysen (Eds.) Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Proc. Sympos. Pure Math., American Mathematical Society, Providence, RI, 2004, in press

[7] Fan, A.H. How many intervals cover a point in Dvoretzky covering ?, Israel J. Math., Volume 131 (2002), pp. 157-184

[8] Holley, R.; Waymire, E.C. Multifractal dimensions and scaling exponents for strongly bounded random fractals, Ann. Appl. Probab., Volume 2 (1992), pp. 819-845

[9] Kahane, J.-P. Sur le chaos multiplicatif, Ann. Sci. Math. Québec, Volume 9 (1985), pp. 105-150

[10] Kahane, J.P. Positive martingales and random measures, Chinese Ann. Math. Ser. B, Volume 8 (1987) no. 1, pp. 1-12

[11] Mandelbrot, B.B. Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées, C. R. Acad. Sci. Paris, Volume 278 (1974), pp. 289-292 (and 355–358)

[12] Molchan, G.M. Scaling exponents and multifractal dimensions for independent random cascades, Commun. Math. Phys., Volume 179 (1996), pp. 681-702

[13] Rajput, B.S.; Rosinski, J. Spectral representations of infinitely divisible processes, Probab. Theory Related Fields, Volume 82 (1989), pp. 451-487