Bernoulli

  • Bernoulli
  • Volume 22, Number 4 (2016), 2579-2608.

Limit theorems for multifractal products of geometric stationary processes

Denis Denisov and Nikolai Leonenko

Full-text: Open access

Abstract

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We present the general conditions for the $\mathcal{L}_{q}$ convergence of cumulative processes to the limiting processes and investigate their $q$th order moments and Rényi functions, which are non-linear, hence displaying the multifractality of the processes as constructed. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios.

Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 2579-2608.

Dates
Received: October 2013
Revised: May 2015
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1462297690

Digital Object Identifier
doi:10.3150/15-BEJ738

Mathematical Reviews number (MathSciNet)
MR3498038

Zentralblatt MATH identifier
1343.60030

Keywords
geometric Gaussian process geometric Ornstein–Uhlenbeck processes Lévy processes log-gamma scenario log-normal scenario log-normal tempered stable scenario long-range dependence log-variance gamma scenario multifractal products multifractal scenarios Rényi function scaling of moments short-range dependence stationary processes superpositions

Citation

Denisov, Denis; Leonenko, Nikolai. Limit theorems for multifractal products of geometric stationary processes. Bernoulli 22 (2016), no. 4, 2579--2608. doi:10.3150/15-BEJ738. https://projecteuclid.org/euclid.bj/1462297690


Export citation

References

  • [1] Anh, V.V. and Leonenko, N.N. (2008). Log-normal, log-gamma and log-negative inverted gamma scenarios in multifractal products of stochastic processes. Statist. Probab. Lett. 78 1274–1282.
  • [2] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2008). Multifractality of products of geometric Ornstein–Uhlenbeck-type processes. Adv. in Appl. Probab. 40 1129–1156.
  • [3] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2008). Log-Euler’s gamma multifractal scenario for products of Ornstein–Uhlenbeck type procseses. Math. Commun. 13 133–148.
  • [4] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2009). Multifractal scaling of products of birth–death processes. Bernoulli 15 508–531.
  • [5] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2009). Multifractal products of stationary diffusion processes. Stoch. Anal. Appl. 27 475–499.
  • [6] Anh, V.V., Leonenko, N.N. and Shieh, N.-R. (2010). Multifractal scenarios for products of geometric Ornstein–Uhlenbeck type processes. In Dependence in Probability and Statistics (P. Doukhan, G. Lang, D. Surgailis and G. Teyssiere, eds.). Lecture Notes in Statist. 200 103–122. Berlin: Springer.
  • [7] Anh, V.V., Leonenko, N.N., Shieh, N.-R. and Taufer, E. (2010). Simulation of multifractal products of Ornstein–Uhlenbeck type processes. Nonlinearity 23 823–843.
  • [8] Bacry, E. and Muzy, J.F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449–475.
  • [9] Barndorff-Nielsen, O.E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2 41–68.
  • [10] Barndorff-Nielsen, O.E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 167–241.
  • [11] Barndorff-Nielsen, O.E. and Shmigel, Y. (2004). Spatio-temporal modeling based on Lévy processes, and its applications to turbulence. Uspekhi Mat. Nauk 59 63–90.
  • [12] Barral, J. and Jin, X. (2014). On exact scaling log-infinitely divisible cascades. Probab. Theory Related Fields 160 521–565.
  • [13] Barral, J., Jin, X. and Mandelbrot, B. (2010). Convergence of complex multiplicative cascades. Ann. Appl. Probab. 20 1219–1252.
  • [14] Barral, J. and Mandelbrot, B.B. (2002). Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 409–430.
  • [15] Barral, J., Peyrière, J. and Wen, Z.-Y. (2009). Dynamics of Mandelbrot cascades. Probab. Theory Related Fields 144 615–631.
  • [16] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
  • [17] Calvet, L. and Fisher, A. (2001). Forecasting multifractal volatility. J. Econometrics 105 27–58.
  • [18] Denisov, D. and Leonenko, N. (2011). Limit theorems for multifractal products of geometric stationary processes. Available at arXiv:1110.2428v2.
  • [19] Denisov, D. and Leonenko, N. (2016). Multifractal scenarios for products of geometric Levy-based stationary models. Stoch. Anal. Appl. To appear.
  • [20] Falconer, K. (1997). Techniques in Fractal Geometry. Chichester: Wiley.
  • [21] Frisch, U. (1995). Turbulence. Cambridge: Cambridge Univ. Press.
  • [22] Harte, D. (2001). Multifractals: Theory and Applications. Boca Raton, FL: Chapman & Hall/CRC.
  • [23] Jaffard, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207–227.
  • [24] Jaffard, S., Abry, P., Roux, S.G., Vedel, B. and Wendt, H. (2010). The contribution of wavelets in multifractal analysis. In Wavelet Methods in Mathematical Analysis and Engineering. Ser. Contemp. Appl. Math. CAM 14 51–98. Beijing: Higher Ed. Press.
  • [25] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 105–150.
  • [26] Kahane, J.-P. (1987). Positive martingales and random measures. Chin. Ann. Math. Ser. B 8 1–12.
  • [27] Karamata, J. (1932). Sur une inégalité relative aux fonctions convexes (in French). Publ. Math. Univ. Belgrade 1 145–148.
  • [28] Kolmogorov, A.N. (1941). Local structure of turbulence in fluid for very large Reynolds numbers. Doklady Acad. Sciences of USSR 31 538–540.
  • [29] Kolmogorov, A.N. (1962). A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13 82–85.
  • [30] Kyprianou, A.E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Berlin: Springer.
  • [31] Leonenko, N.N. and Shieh, N.-R. (2013). Rényi function for multifractal random fields. Fractals 21 1350009, 13.
  • [32] Ludeña, C. (2008). $L^{p}$-variations for multifractal fractional random walks. Ann. Appl. Probab. 18 1138–1163.
  • [33] Mandelbrot, B.B. (1997). Fractals and Scaling in Finance. Selected Works of Benoit B. Mandelbrot. New York: Springer.
  • [34] Mannersalo, P., Norros, I. and Riedi, R.H. (2002). Multifractal products of stochastic processes: Construction and some basic properties. Adv. in Appl. Probab. 34 888–903.
  • [35] Matsui, M. and Shieh, N.-R. (2009). On the exponentials of fractional Ornstein–Uhlenbeck processes. Electron. J. Probab. 14 594–611.
  • [36] Mörters, P. and Shieh, N.-R. (2004). On the multifractal spectrum of the branching measure on a Galton–Watson tree. J. Appl. Probab. 41 1223–1229.
  • [37] Muzy, J.-F. and Bacry, E. (2002). Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws. Phys. Rev. E (3) 66 056121.
  • [38] Novikov, E.A. (1994). Infinitely divisible distributions in turbulence. Phys. Rev. E (3) 50 R3303–R3305.
  • [39] Riedi, R.H. (2003). Multifractal processes. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M. Taqqu, eds.) 625–716. Boston, MA: Birkhäuser.
  • [40] Schertzer, D., Lovejoy, S., Schmitt, F., Chigirinskaya, Y. and Marsan, D. (1997). Multifractal cascade dynamics and turbulent intermittency. Fractals 5 427–471.
  • [41] Schmitt, F.G. (2003). A causal multifractal stochastic equation and its statistical properties. Eur. Phys. J. B 34 85–98.
  • [42] Shieh, N.-R. and Taylor, S.J. (2002). Multifractal spectra of branching measure on a Galton–Watson tree. J. Appl. Probab. 39 100–111.