Bernoulli

  • Bernoulli
  • Volume 17, Number 2 (2011), 749-780.

Measuring the roughness of random paths by increment ratios

Jean-Marc Bardet and Donatas Surgailis

Full-text: Open access

Abstract

A statistic based on increment ratios (IR’s) and related to zero crossings of an increment sequence is defined and studied for the purposes of measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (which we shall call the IR-roughness) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. First, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Second, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Third, the IR-roughness of a Lévy process with an $α$-stable tangent process is established and can be used to estimate the fractional parameter $α ∈ (0, 2)$ following a central limit theorem.

Article information

Source
Bernoulli, Volume 17, Number 2 (2011), 749-780.

Dates
First available in Project Euclid: 5 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1302009246

Digital Object Identifier
doi:10.3150/10-BEJ291

Mathematical Reviews number (MathSciNet)
MR2787614

Zentralblatt MATH identifier
1248.60042

Keywords
diffusion processes estimation of the local regularity function of stochastic process fractional Brownian motion Hölder exponent Lévy processes limit theorems multifractional Brownian motion tangent process zero crossings

Citation

Bardet, Jean-Marc; Surgailis, Donatas. Measuring the roughness of random paths by increment ratios. Bernoulli 17 (2011), no. 2, 749--780. doi:10.3150/10-BEJ291. https://projecteuclid.org/euclid.bj/1302009246


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References

  • [1] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency financial data. Ann. Statist. 37 2202–2244.
  • [2] Arcones, M.A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242–2274.
  • [3] Ayache, A., Benassi, A., Cohen, S. and Lévy Véhel, J. (2005). Regularity and identification of generalized multifractional Gaussian processes. In Séminare de Probabilités XXXVIII. Lecture Notes in Math. 1857 290–312. Berlin: Springer.
  • [4] Bardet, J.-M. and Bertrand, P. (2007). Definition, properties and wavelet analysis of multiscale fractional Brownian motion. Fractals 15 73–87.
  • [5] Bardet, J.-M. and Surgailis, D. (2009). A central limit theorem for triangular arrays of nonlinear functionals of Gaussian vectors. Preprint.
  • [6] Bardet, J.-M. and Surgailis, D. (2010). Nonparametric estimation of the local Hurst function of multifractional processes. Preprint.
  • [7] Belomestny, D. (2010). Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 317–351.
  • [8] Benassi, A., Cohen, S. and Istas, J. (1998). Identifying the multifractal function of a Gaussian process. Statist. Probab. Lett. 39 337–345.
  • [9] Benassi, A., Jaffard, S. and Roux, D. (1997). Gaussian processes and pseudodifferential elliptic operators. Rev. Mat. Iberoamericana 13 19–89.
  • [10] Berk, K.N. (1973). A central limit theorem for m-dependent random variables with unbounded m. Ann. Probab. 1 352–354.
  • [11] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
  • [12] Coeurjolly, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 199–227.
  • [13] Coeurjolly, J.-F. (2005). Identification of multifractional Brownian motion. Bernoulli 11 987–1008.
  • [14] Coeurjolly, J.-F. (2007). Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Ann. Statist. 36 1404–1434.
  • [15] Cramér, H. and Leadbetter, M.R. (1967). Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications. New York: Wiley.
  • [16] Csörgő, M. and Mielniczuk, J. (1996). The empirical process of a short-range dependent stationary sequence under Gaussian subordination. Probab. Theory Related Fields 104 15–25.
  • [17] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749–1766.
  • [18] Dobrushin, R.L. (1980). Automodel generalized random fields and their renorm group. In Multicomponent Random Systems (R.L. Dobrushin and Y.G. Sinai, eds.) 153–198. New York: Dekker.
  • [19] Falconer, K. (2002). Tangent fields and the local structure of random fields. J. Theoret. Probab. 15 731–750.
  • [20] Falconer, K. (2003). The local structure of random processes. J. London Math. Soc. 67 657–672.
  • [21] Feuerverger, A., Hall, P. and Wood, A.T.A. (1994). Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings. J. Time Series Anal. 15 587–606.
  • [22] Fox, R. and Taqqu, M.S. (1986). Large-sample properties of parameter estimates for strongly dependent Gaussian time series. Ann. Statist. 14 517–532.
  • [23] Gikhman, I.I. and Skorohod, A.V. (1969). Introduction to the Theory of Random Processes. Philadelphia: Saunders.
  • [24] Gugushvili, S. (2008). Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process. J. Nonparametr. Stat. 21 321–343.
  • [25] Guyon, X. and Leon, J. (1989). Convergence en loi des H-variations d’un processus gaussien stationnaire. Ann. Inst. H. Poincaré Probab. Statist. 25 265–282.
  • [26] Hall, P. and Wood, A. (1993). On the performance of box-counting estimators of fractal dimension. Biometrika 80 246–252.
  • [27] Ho, H.-C. and Sun, T.C. (1987). A central limit theorem for non-instantaneous filters of a stationary Gaussian process. J. Multivariate Anal. 22 144–155.
  • [28] Ibragimov, I.A. and Linnik. Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff.
  • [29] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407–436.
  • [30] Neumann, M. and Reiß, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 223–248.
  • [31] Peltier, R. and Lévy-Vehel, J. (1994). A new method for estimating the parameter of fractional Brownian motion, INRIA. Preprint. Available at http://hal.inria.fr/docs/00/07/42/79/PDF/RR-2396.pdf.
  • [32] Philippe, A., Surgailis, D. and Viano, M.-C. (2006). Invariance principle for a class of non stationary processes with long memory. C. R. Acad. Sci. Paris Ser. 1 342 269–274.
  • [33] Philippe, A., Surgailis, D. and Viano, M.-C. (2008). Time-varying fractionally integrated processes with nonstationary long memory. Theory Probab. Appl. 52 651–673.
  • [34] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ. Press.
  • [35] Shilov, G.E. and Gurevich, B.L. (1967). Integral, Measure and Derivative (in Russian). Moscow: Nauka.
  • [36] Stoncelis, M. and Vaičiulis, M. (2008). Numerical approximation of some infinite Gaussian series and integrals. Nonlinear Anal. Modell. Control 13 397–415.
  • [37] Surgailis, D. (2008). Nonhomogeneous fractional integration and multifractional processes. Stochastic Process. Appl. 118 171–198.
  • [38] Surgailis, D., Teyssière, G. and Vaičiulis, M. (2008). The increment ratio statistic. J. Multivariate Anal. 99 510–541.
  • [39] Taqqu, M.S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrsch. Verw. Gebiete 40 203–238.
  • [40] Vaičiulis, M. (2009). An estimator of the tail index based on increment ratio statistics. Lithuanian Math. J. 49 222–233.
  • [41] Van der Vaart, A. (1998). Asymptotic Statistics. Cambridge: Cambridge Univ. Press.