Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-29T18:39:08.872Z Has data issue: false hasContentIssue false

Superposition of Diffusions with Linear Generator and its Multifractal Limit Process

Published online by Cambridge University Press:  15 May 2003

Endre Iglói
Institute of Mathematics and Informatics, University of Debrecen, 4010 Debrecen, PF 12, Hungary;
György Terdik
Institute of Mathematics and Informatics, University of Debrecen, 4010 Debrecen, PF 12, Hungary;
Get access


In this paper a new multifractal stochastic process called Limit of the Integrated Superposition of Diffusion processes with Linear differencial Generator (LISDLG) is presented which realistically characterizes the network traffic multifractality. Several properties of the LISDLG model are presented including long range dependence, cumulants, logarithm of the characteristic function, dilative stability, spectrum and bispectrum. The model captures higher-order statistics by the cumulants. The relevance and validation of the proposed model are demonstrated by real data of Internet traffic.

Research Article
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


O.E. Barndorff-Nielsen, Superposition of Ornstein-Uhlenbeck type processes, Technical Report 1999-2, MaPhySto. Aarhus University (1999), URL:
J. Beran, Statistics for Long-Memory Processes. Chapman & Hall, Monogr. Statist. Appl. Probab. 61 (1994).
Brillinger, D.R., An introduction to polyspectra. Ann. Math. Statist. 36 (1965) 1351-1374. CrossRef
Calvet, L. and Fisher, A., Forecasting multifractal volatility. J. Econometrics 105 (2001) 27-58. CrossRef
P. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electron. Commun. Probab. 3 (1998) 95-107.
P. Carmona, L. Coutin and G. Montseny, Applications of a representation of long-memory Gaussian processes. Stochastic Process. Appl. (submitted), URL:
Clifford, P. and Wei, G., The equivalence of the Cox process with squared radial Ornstein-Uhlenbeck intensity and the death process in a simple population model. Ann. Appl. Probab. 3 (1993) 863-873. CrossRef
Cox, D.R., Long-range dependence, non-linearity and time irreversibility. J. Time Ser. Anal. 12 (1991) 329-335. CrossRef
Cox, J.C., Ingersoll, J.E. and Ross, S.A., A theory of the term structure of interest rates. Econometrica 53 (1985) 385-407. CrossRef
Dankel Jr, T.., On the distribution of the integrated square of the Ornstein-Uhlenbeck process. SIAM J. Appl. Math. 51 (1991) 568-574. CrossRef
E.B. Dynkin, Markov Processes, Vols. 1-2. Springer-Verlag, Berlin-Göttingen-Heidelberg (1965).
Feller, W., Two singular diffusion problems. Ann. Math. (2) 54 (1951) 173-182. CrossRef
Granger, C., Long-memory relationship and aggregation of dynamic models. J. Econometrics 14 (1980) 227-238. CrossRef
Griffiths, R.C., Infinitely divisible multivariate gamma distributions. Sankhya Ser. A 32 (1970) 393-404.
E. Iglói, Long-range dependent processes with real bispectrum are third order nonlinear, Technical Report 259 (2001/1). University of Debrecen, Institute of Mathematics and Informatics (2001).
Iglói, E. and Terdik, G., Long-range dependence through Gamma-mixed Ornstein-Uhlenbeck process. Electron. J. Probab. (EJP) 4 (1999) 1-33.
N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes. North-Holland Publishing Co.,Amsterdam (1981).
Jaffard, S., The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 (1999) 207-227. CrossRef
Karlin, S. and McGregor, J., Classical diffusion processes and total positivity. J. Math. Anal. Appl. 1 (1960) 163-183. CrossRef
Kawazu, K. and Watanabe, S., Branching processes with immigration and related limit theorems. Teor. Verojatnost. Primenen. 16 (1971) 34-51.
Kolmogorov, A.N., Über die analitischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931) 415-458. CrossRef
Kolmogorov, A.N., Wiener spiral and some other interesting curve in Hilbert space. C. R. Acad. Sci. URSS 26 (1940) 115-118.
Kolmogorov, A.N., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS 30 (1941) 301-305.
Krishnaiah, P.R. and Rao, M.M., Remarks on a multivariate gamma distribution. Amer. Math. Monthly 68 (1961) 342-346. CrossRef
Krishnamoorthy, A.S. and Parthasarathy, M., A multi-variate gamma-type distribution. Ann. Math. Statist. 22 (1951) 549-557. CrossRef
Leland, W.E., Taqqu, M.S., Willinger, W. and Wilson, D.W., On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 2 (1994) 1-15. CrossRef
Lowen, S.B. and Teich, M.C., Fractal renewal processes generate 1/f noise. Phys. Rev. E 47 (1993) 992-1001. CrossRef
Mandelbrot, B.B., Long-run linearity, locally Gaussian processes, h-spectra and infinite variances. Technometrics 10 (1969) 82-113.
Mandelbrot, B.B. and Van Ness, J.W., Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422-437. CrossRef
A.P. Prudnikov, Y.A. Britshkov and O.I. Maritshev, Integrali i ryadi. Nauka, Moscow. Appl. Math. (1981), in Russian.
Riedi, H.R., Crouse, S.M., Ribeiro, J.V. and Baraniuk, G.R., A multifractal wavelet model with application to network traffic. IEEE Trans. Inform. Theory 45 (1999) 992-1018. CrossRef
Ryu, B. and Lowen, S.B., Point process models for self-similar network traffic, with applications. Comm. Statist. Stochastic Models 14 (1998) 735-761. CrossRef
G. Samorodnitsky and M.S. Taqqu, Linear models with long-range dependence and finite or infinite variance, edited by D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt and M.S. Taqqu, New Directions in Time Series Analysis, Part II. Springer-Verlag, New York, IMA Vol. Math. Appl. 46 (1992) 325-340.
Shiga, T. and Watanabe, S., Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 (1973) 37-46. CrossRef
A.N. Shiryaev, Probability. Springer-Verlag, New York (1996).
S.E. Shreve, Steven shreve's lectures on stochastic calculus and finance, Online:
Sinai, Y.G., Self-similar probability distributions. Theor. Probab. Appl. 21 (1976) 64-84. CrossRef
G. Szego, Orthogonal polynomials, in Colloquium Publ., Vol. XXIII. American Math. Soc., New York (1936).
M.S. Taqqu, V. Teverovsky and W. Willinger, Is network traffic self-similar or multifractal? Fractals 5 (1997) 63-73.
G. Terdik, Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis; A Frequency Domain Approach. Springer-Verlag, New York, Lecture Notes in Statist. 142 (1999).
Terdik, G., Gál, Z., Molnár, S. and Iglói, E., Bispectral analysis of traffic in high speed networks. Comput. Math. Appl. 43 (2002) 1575-1583. CrossRef
Willinger, W., Taqqu, M.S., Sherman, R. and Wilson, D.V., Self-similarity through high variability: Statistical analysis of Ethernet LAN traffic at the source level. Comput. Commun. Rev. 25 (1995) 100-113. CrossRef
Yamada, T. and Watanabe, S., On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 (1971) 155-167. CrossRef