## The Annals of Applied Probability

### On sampling of stationary increment processes

J. M. P. Albin

#### Abstract

Under a complex technical condition, similar to such used in extreme value theory, we find the rate q(ɛ)−1 at which a stochastic process with stationary increments ξ should be sampled, for the sampled process ξ(⌊⋅/q(ɛ)⌋q(ɛ)) to deviate from ξ by at most ɛ, with a given probability, asymptotically as ɛ↓0. The canonical application is to discretization errors in computer simulation of stochastic processes.

#### Article information

Source
Ann. Appl. Probab., Volume 14, Number 4 (2004), 2016-2037.

Dates
First available in Project Euclid: 5 November 2004

https://projecteuclid.org/euclid.aoap/1099674087

Digital Object Identifier
doi:10.1214/105051604000000468

Mathematical Reviews number (MathSciNet)
MR2099661

Zentralblatt MATH identifier
1075.60023

#### Citation

Albin, J. M. P. On sampling of stationary increment processes. Ann. Appl. Probab. 14 (2004), no. 4, 2016--2037. doi:10.1214/105051604000000468. https://projecteuclid.org/euclid.aoap/1099674087

#### References

• Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Probab. 18 92--128.
• Albin, J. M. P. (1992). On the general law of iterated logarithm with application to selfsimilar processes and to Gaussian processes in $\mathbbR^n$ and Hilbert space. Stochastic Process. Appl. 41 1--31.
• Albin, J. M. P. (1993). Extremes of totally skewed stable motion. Statist. Probab. Lett. 16 219--224.
• Albin, J. M. P. (1998). Extremal theory for self-similar processes. Ann. Probab. 26 743--793.
• Albin, J. M. P. (1999). Extremes of totally skewed $\alpha$-stable processes. Stochastic Process. Appl. 79 185--212.
• Albin, J. M. P. (2003a). On extremes of infinitely divisible Ornstein--Uhlenbeck processes. Available at www.math.chalmers.se/~palbin/ornstein.ps.
• Albin, J. M. P. (2003b). Large deviations of stationary infinitely divisible processes. Preprint. Available at www.math.chalmers.se/~palbin/id.ps.
• Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.
• Belyaev, Yu. K. and Simonyan, A. H. (1979). Asymptotic properties of deviations of a sample path of a Gaussian process from approximation by broken line for decreasing width of quantization. In Random Processes and Fields (Yu. K. Belyaev, ed.) 9--21. Moscow Univ. Press. (In Russian.)
• Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Probab. 10 1--46.
• Berman, S. M. (1986). The supremum of a process with stationary independent and symmetric increments. Stochastic Process. Appl. 23 281--290.
• Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Belmont, CA.
• Hüsler, J. (1999). Extremes of Gaussian processes, on results of Piterbarg and Seleznjev. Statist. Probab. Lett. 44 251--258.
• Kuelbs, J. and Li, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133--157.
• Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
• Marcus, M. B. (2000). Probability estimates for lower levels of certain Gaussian processes with stationary increments. In High Dimensional Probability (E. Giné, D. M. Mason and J. A. Wellner, eds.) 173--179. Birkhäuser, Boston.
• Pickands, J., III (1969a). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51--73.
• Pickands, J., III (1969b). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145 75--86.
• Piterbarg, V. I. and Seleznjev, O. (1994). Linear interpolation of random processes and extremes of a sequence of Gaussian nonstationary processes. Technical Report 446, Dept. Statistics, Univ. North Carolina, Chapel Hill.
• Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, London.
• Scheffé, H. (1947). A useful convergence theorem for probability distributions. Ann. Math. Statist. 18 434--438.
• Seleznjev, O. (1996). Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments. Adv. in Appl. Probab. 28 481--499.
• Willekens, E. (1987). On the supremum of an infinitely divisible process. Stochastic Process. Appl. 26 173--175.