Bernoulli

  • Bernoulli
  • Volume 15, Number 2 (2009), 508-531.

Multifractal scaling of products of birth–death processes

Vo V. Anh, Nikolai N. Leonenko, and Narn-Rueih Shieh

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Abstract

We investigate the scaling properties of products of the exponential of birth–death processes with certain given marginal discrete distributions and covariance structures. The conditions on the mean, variance and covariance functions of the resulting cumulative processes are interpreted in terms of the moment generating functions. We provide four illustrative examples of Poisson, Pascal, binomial and hypergeometric distributions. We establish the corresponding log-Poisson, log-Pascal, log-binomial and log-hypergeometric scenarios for the limiting processes, including their Rényi functions and dependence properties.

Article information

Source
Bernoulli, Volume 15, Number 2 (2009), 508-531.

Dates
First available in Project Euclid: 4 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1241444900

Digital Object Identifier
doi:10.3150/08-BEJ156

Mathematical Reviews number (MathSciNet)
MR2543872

Zentralblatt MATH identifier
1215.60048

Keywords
geometric birth–death processes log-binomial scenario log-Pascal scenario log-Poisson scenario multifractal products

Citation

Anh, Vo V.; Leonenko, Nikolai N.; Shieh, Narn-Rueih. Multifractal scaling of products of birth–death processes. Bernoulli 15 (2009), no. 2, 508--531. doi:10.3150/08-BEJ156. https://projecteuclid.org/euclid.bj/1241444900


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References

  • [1] Abramowitz, M. and Stegun, I.A., eds. (1972)., Handbook of Mathematical Functions. New York: Dover.
  • [2] Anh, V.V., Lau, K.-S. and Yu, Z.-G. (2001). Multifractal characterization of complete genomes, J. Physics A Math. General 34 7127–7139.
  • [3] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2008). Multifractality of products of geometric Ornstein–Uhlenbeck type processes., Adv. Appl. Probab. To appear.
  • [4] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2008). Multifractal products of stationary diffusion processes., Stoch. Anal. Appl. (To appear.).
  • [5] Barral, J. andMandelbrot, B. (2002). Multiplicative products of cylindrical pulses., Probab. Theory Related Fields 124 409–430.
  • [6] Bhattacharya, R. and Waymire, E. (1992)., Stochastic Processes and Applications. New York: Wiley.
  • [7] Chihara, T.S. (1978)., An Introduction to Orthogonal Polynomials. New York: Gordon and Breach.
  • [8] Dubrulle, B. (1994). Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance., Phys. Rev. Lett. 73 959–996.
  • [9] Dynkin, E.B. and Yushkevich, A.A. (1969)., Markov Processes: Theorems and Problems. New York: Plenum Press.
  • [10] Falconer, K. (1997)., Techniques in Fractal Geometry. Chichester: Wiley.
  • [11] Frisch, U. (1995)., Turbulence. Cambridge: Cambridge Univ. Press.
  • [12] Frisch, U. and Parisi, G. (1985). On the singularity structure of fully developed turbulence., International School of Physics “Enrico Fermi” Course 88 84–88. North Holland.
  • [13] Halsey, T., Jensen, M., Kadanoff, L., Procaccia, I. and Shraiman, B. (1986). Fractal measures and their singularities: The characterization of strange sets., Phys. Rev. A 33 1141–1151.
  • [14] Hentschel, H. and Procaccia, I. (1983). The infinite number of generalized dimensions of fractals and strange attractors., Physica 80 435–444.
  • [15] Jaffard, S. (1999). The multifractal nature of Lévy processes., Probab. Theory Related Fields 114 207–227.
  • [16] Kahane, J.-P. (1985). Sur la chaos multiplicatif., Ann. Sc. Math. Québec 9 105–150.
  • [17] Kahane, J.-P. (1987). Positive martingale and random measures., Chinese Ann. Math. 8B 1–12.
  • [18] Karlin, S. and McGregor, J.L. (1957). The classification of birth–death processes., Trans. Amer. Math. Soc. 86 366–400.
  • [19] Karlin, S. and Taylor, H. (1977)., A First Course in Stochastic Processes. Academic Press.
  • [20] Kolmogorov, A.N. (1962). On refinement of previous hypotheses concerning the local structure in viscous incompressible fluid at high Reynolds number., J. Fluid Mech. 13 82–85.
  • [21] Lau, K.S. (1999). Iterated function systems with overlaps and multifractal structure. In, Trends in Probability and Related Analysis (N. Kono and N.-R. Shieh, eds.) 35–76. River Edge, NJ: World Scientific.
  • [22] Mannersalo, P., Norros, I. and Riedi, R. (2002). Multifractal products of stochastic processes: Construction and some basic properties., Adv. Appl. Probab. 34 888–903.
  • [23] Molchan, M. (1996). Scaling exponents and multifractal dimensions for independent random cascades., Comm. Math. Phys. 179 681–702.
  • [24] Mörters, P. and Shieh, N.-R. (2002). Thin and thick points for branching measure on a Galton–Watson tree., Statist. Probab. Lett. 58 13-22.
  • [25] Mörters, P. and Shieh, N.-R. (2004). On multifractal spectrum of the branching measure on a Galton–Watson tree, J. Appl. Probab. 41 1223-1229.
  • [26] Mörters, P. and Shieh, N.-R. (2008). Multifractal analysis of branching measure on a Galton–Watson tree. In, Proc. 3rd International Congress of Chinese Mathematicians, Hong Kong (K.S. Lau, Z.P. Xin and S.T. Yau, eds.). AMS/IP Studies in Advanced Mathematics 42 655–662.
  • [27] Novikov, E.A. (1994). Infinitely divisible distributions in turbulence., Phys. Rev. E 50 R3303–R3305.
  • [28] Riedi, R. (2003). Multifractal processes. In, Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M. Taqqu, eds.) 625–716. Boston: Birkhäuser.
  • [29] Schoutens, W. (2000)., Stochastic Processes and Orthogonal Polinomials. New York: Springer.
  • [30] She, Z.S. and Lévêque, E. (1994). Universal scaling laws in fully developed turbulence., Phys. Rev. Lett. 72 336–339.
  • [31] She, Z.S. and Waymire, E.C. (1995). Quantized energy cascade and log-Poisson statistics in fully developed turbulence., Phys. Rev. Lett. 74 262–265.
  • [32] Shieh, N.-R. and Taylor, S.J. (2002). Multifractal spectra of branching measure on a Galton–Watson tree., J. Appl. Probab. 39 100–111.
  • [33] Taylor, S. J. (1995). Super Brownian motion is a fractal measure for which the multifractal formalism is invalid., Fractals 3 737–746.
  • [34] Waymire, E. and Williams, S. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades., Trans. Amer. Math. Soc. 348 585–632.
  • [35] Waymire, E. and Williams, S. (1997). Markov cascades. In, Classical and Modern Branching Processes (R.B. Athreya and P. Jagers, eds.) 305–321. New York: Springer.