Abstract
A function \(f=f_T\) is called least energy approximation to a function B on the interval [0, T] with penalty Q if it solves the variational problem
For quadratic penalty, the least energy approximation can be found explicitly. If B is a random process with stationary increments, then on large intervals, \(f_T\) also is close to a process of the same class, and the relation between the corresponding spectral measures can be found. We show that in a long run (when \(T\rightarrow \infty \)), the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and Lévy processes, we complete this result with almost sure and \(L^1\) convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion.
Similar content being viewed by others
References
Davies, P.L., Kovac, A.: Local extremes, runs, strings and multiresolution. Ann. Stat. 29(1), 1–65 (2001)
Embrechts, P., Maejima, M.: Selfsimilar Processes. Princeton University Press, Princeton (2002)
Grenander, U.: Stochastic processes and statistical inference. Arkiv Mat. 1, 195–277 (1950)
Karatzas, I.: On a stochastic representation for the principal eigenvalue of a second order differential equation. Stochastics 3, 305–321 (1980)
Lifshits, M., Setterqvist, E.: Energy of taut string accompanying Wiener process. Stoch. Process. Their Appl. 125, 401–427 (2015)
Mammen, E., van de Geer, S.: Locally adaptive regression splines. Ann. Stat. 25(1), 387–413 (1997)
Maruyama, G.: The harmonic analysis of stationary stochastic processes. Mem. Fac. Sci. Kyusyu Univ. A4, 45–106 (1949)
Rosinski, J., Zak, T.: Simple conditions for mixing of infinitely divisible processes. Stoch. Process. Their Appl. 61, 277–288 (1996)
Rosinski, J., Zak, T.: The equivalence of ergodicity and weak mixing for infinitely divisible processes. J. Theor. Probab. 10, 73–86 (1997)
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)
Scherzer, O., et al.: Variational Methods in Imaging. Applied Mathematical Sciences, vol. 167. Springer, New York (2009)
Setterqvist, E., Forchheimer, R.: Application of \(\varphi \)-stable sets to a buffered real-time communication system. In: Proceedings of the 10th Swedish National computer networking workshop (2014)
Yaglom, A.M.: An Introduction to the Theory of Stationary Random Functions. Prentice Hall Inc., Englewood Cliffs (1962). (revised English edition)
Acknowledgments
The work of Mikhail Lifshits was supported by Grants NSh.2504.2014.1, RFBR 13-01-00172, and SPbSU 6.38.672.2013.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Kabluchko, Z., Lifshits, M. Least Energy Approximation for Processes with Stationary Increments. J Theor Probab 30, 268–296 (2017). https://doi.org/10.1007/s10959-015-0642-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-015-0642-8
Keywords
- Least energy approximation
- Gaussian process
- Lévy process
- Fractional Brownian motion
- Process with stationary increments
- Taut string
- Variational calculus