Abstract
We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to certain important families of processes that are not self-similar in the conventional sense. This includes Hougaard Lévy processes such as the Poisson processes, Brownian motions with drift and the inverse Gaussian processes, and some new fractional Hougaard motions defined as moving averages of Hougaard Lévy process. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.
Similar content being viewed by others
References
O.E. Barndorff-Nielsen and J. Schmiegel, Time change, volatility and turbulence, in A. Sarychev, M. Guerra A. Shiryaev, and M. do R. Grossinho (Eds.), Mathematical Control Theory and Finance, Springer, Berlin, 2008, pp. 29–53.
O.E. Barndorff-Nielsen and S. Thorbjørnsen, Classical and free infinite divisibility and Lévy processes, in M. Schüermann and U. Franz (Eds.), Quantum Independent Increment Processes II, Lect. Notes Math., Vol. 1866, Springer-Verlag, Berlin, Heidelberg, 2006, pp. 33–159.
J. Beran, Statistics for Long-Memory Processes, Chapman & Hall, London, 1994.
N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge, 1987.
P. Embrechts and M. Maejima, Selfsimilar Processes, Princeton Univ. Press, Princeton, NJ, 2002.
P. Hougaard, M.-L.T. Lee, and G.A. Whitmore, Analysis of overdispersed count data by mixtures of Poisson variables and Poisson processes, Biometrics, 53:1225–1238, 1997.
G. Jona-Lasinio, The renormalization group: A probabilistic view, Nuovo Cimento, 26:99–119, 1975.
B. Jørgensen, Exponential dispersion models and extensions: A review, Int. Stat. Rev., 60:5–20, 1992.
B. Jørgensen, The Theory of Dispersion Models, Chapman & Hall, London, 1997.
B. Jørgensen and J.R. Martínez, The Lévy–Khinchine representation of the Tweedie family, Braz. J. Probab. Stat., 10:225–233, 1996.
U. Küchler and M. Sørensen, Exponential Families of Stochastic Processes, Springer-Verlag, New York, 1997.
J. Lamperti, Semi-stable stochastic processes, Trans. Am. Math. Soc., 104:62–78, 1962.
M.-L.T. Lee and G.A. Whitmore, Stochastic processes directed by randomized time, J. Appl. Probab., 30:302–314, 1993.
B.B. Mandelbrot and J.W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10:422–437, 1968.
T. Marquardt, Fractional Lévy processes with an application to long memory moving average processes, Bernoulli, 12:1099–1126, 2006.
B. Rajput and J. Rosiński, Spectral representation of infinitely divisible processes, Probab. Theory Relat. Fields, 82:451–487, 1989.
M.S. Taqqu, Fractional Brownian motion and long-range dependence, in P. Doukhan, G. Oppenheim, and M.S. Taqqu (Eds.), Theory and Applications of Long-Range Dependence, Birkhäuser, Boston, 2003, pp. 5–38.
V. Vinogradov, Properties of certain Lévy and geometric Lévy processes, Commun. Stoch. Anal., 2:193–208, 2008.
M.T. Wasan, On an inverse Gaussian process, Skand. Aktuarietidskr., 51:69–96, 1968.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jørgensen, B., Martínez, J.R. & Demétrio, C.G. Self-similarity and Lamperti convergence for families of stochastic processes. Lith Math J 51, 342–361 (2011). https://doi.org/10.1007/s10986-011-9131-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-011-9131-7