Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T13:24:17.195Z Has data issue: false hasContentIssue false
  • English
  • Français

A Gamma Activity Time Process with Noninteger Parameter and Self-Similar Limit

Part of: Stochastic processes Applications

Published online by Cambridge University Press:  14 July 2016

Richard Finlay  and
Eugene Seneta
Richard Finlay*
Affiliation:
University of Sydney
Eugene Seneta*
Affiliation:
University of Sydney
*
∗Postal address: School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia.
∗Postal address: School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct a process with gamma increments, which has a given convex autocorrelation function and asymptotically a self-similar limit. This construction validates the use of long-range dependent t and variance-gamma subordinator models for actual financial data as advocated in Heyde and Leonenko (2005) and Finlay and Seneta (2006), in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting.

Keywords

Gamma processvariance-gamma distributiont distributionsubordinator modelconstructionlong-range dependenceself-similarity

MSC classification

Secondary: 60G10: Stationary processes 60G18: Self-similar processes 62P20: Applications to economics
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 950 - 959
Copyright
Copyright © Applied Probability Trust 2007 

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Finlay, R. and Seneta, E. (2006). Stationary-increment Student and variance-gamma processes. J. Appl. Prob. 43, 441–453. (Correction: 1207.)Google Scholar
Gneiting, T. (2000). Power-law correlations, related models for long-range dependence and their simulation. J. Appl. Prob. 37, 1104–1109.Google Scholar
Heyde, C. C. and Leonenko, N. N. (2005). Student processes. Adv. Appl. Prob. 37, 342–365.CrossRefGoogle Scholar
Liu, M. (2000). Modeling long memory in stock market volatility. J. Econometrics 99, 139–171.Google Scholar
Mukherjea, A., Rao, M. and Suen, S. (2006). A note on moment generating functions. Statist. Prob. Lett. 76, 1185–1189.CrossRefGoogle Scholar
Sly, A. (2006). Self-similarity, multifractionality and multifractality. , Australian National University.Google Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.Google Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.CrossRefGoogle Scholar
Taqqu, M. S. and Levy, J. B. (1986). Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics: A Survey of Recent Results, eds Eberlein, E. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 73–89.CrossRefGoogle Scholar