Open Access Open Access  Restricted Access Subscription or Fee Access

Probability Characteristics of the Absolute Maximum of the Discontinuous Homogeneous Gaussian Random Field


(*) Corresponding author


Authors' affiliations


DOI: https://doi.org/10.15866/iremos.v11i5.15019

Abstract


A new method is introduced in order to determine the limiting characteristics of the homogeneous Gaussian random field with the non-differentiable correlation function at zero. This method is based on the representation of a field in the small neighborhood of the maximum point by the superposition of the statistically independent locally Markov Gaussian random processes. Then, using the expressions for the distribution function of the Markov Gaussian process, it is possible to obtain the asymptotic approximations for both the probability of the threshold crossing by the absolute maximum of the homogeneous Gaussian random field and the distribution function of the absolute maximum of the field. It is shown that the proposed approximations are monotonic and valid in a much wider range of both threshold values and the area of the definitional domain of the random field, unlike the commonly known approximations. The applicability borders of the introduced theoretical formulas are established by means of the statistical computer simulation.
Copyright © 2018 Praise Worthy Prize - All rights reserved.

Keywords


Discontinuous Random Field; Distribution Function of the Absolute Maximum; Local Markov Approximation; Statistical Simulation

Full Text:

PDF


References


A. A. Sveshnikov, Applied Methods of the Random Functions Theory [in Russian] (Nauka, 1968).
http://dx.doi.org/10.2307/3612758

E. J. Gumbel, Statistics of Extremes (Columbia University Press, 1958).
http://dx.doi.org/10.1080/00137916108928620

M. R. Leadbetter, G. Lindgren, H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer, 1983).
http://dx.doi.org/10.1007/978-1-4612-5449-2

Yu. K. Belyaev, Distribution of the Maximum of a Random Field and Its Application to Reliability Problems, Engineering Cybernetics, (Issue 2):269-276, February 1970.

A. P. Khusu, Yu. R. Vitenberg, V. A. Palmov, Roughness of Surfaces [in Russian] (Nauka, 1975).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 3. Elements of Random Fields (Springer-Verlag, 1989).

V. A. Svetlitskiy, Random Vibrations of Mechanical Systems [in Russian] (Mashinostroenie, 1991).

S. A. Akhmanov, Yu. E. Dyakov, A. S. Chirkin, Introduction to the Statistical Radio Physics and Optics [in Russian] (Nauka, 1981).

M. P. Dolukhanov, Fluctuation Processes on Radio Wave Propagation [in Russian] (Svyaz’, 1991).

V. I. Tikhonov, Optimal Signal Reception [in Russian] (Radio i Svyaz’, 1983).

H. L. van Trees, K. L. Bell, Z. Tian, Detection, Estimation, and Modulation Theory, Part I, Detection, Estimation, and Filtering Theory (Wiley, 2013).

V. I. Tikhonov, Outliers of Random Processes [in Russian] (Nauka, 1970).

V. I. Tikhonov, V. I. Khimenko, Outliers of Random Process Trajectories [in Russian] (Nauka, 1987).

Random Processes. Sample Functions and Crossings [in Russian], Edited by Yu. K. Belyaev (Mir, 1978).

J. A. Abrahams, Survey of Recent Progress on Level-Crossing Problems for Random Process (Springer-Verlag, 1986).
http://dx.doi.org/10.1007/978-1-4612-4904-7_2

O. V. Chernoyarov, A. V. Salnikova, and Ya. A. Kupriyanova, Definition of Probability Characteristics of the Absolute Maximum of Non-Gaussian Random Processes by Example of Hoyt Process, American Journal of Theoretical and Applied Statistics, Vol. 2 (Issue 3):54-60, May 2013.
http://dx.doi.org/10.11648/j.ajtas.20130203.13

O. V. Chernoyarov, A. V. Salnikova, L. A. Golpaiegani, On probability of the Gaussian random processes crossing the barriers, 3rd International Conference on Frontiers of Signal Processing, pp. 1-7, Paris, France, September 2017.
http://dx.doi.org/10.1109/icfsp.2017.8097050

H. Cramer, M. Leandbetter, Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications (Wiley, 1967).


V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields (American Mathematical Society, 1996).
http://dx.doi.org/10.1090/mmono/148

S. O. Rise, Mathematical Analysis of Random Noise, Bell System Technical Journal, Vol. 24 (Issue 1):46-156, January 1945.
http://dx.doi.org/10.1002/j.1538-7305.1945.tb00453.x

I. A. Ibragimov, R. Z. Has’minskii, Statistical Estimation–Asymptotic Theory (Springer, 1981).

V. I. Tikhonov, Statistical Radio Engineering [in Russian] (Radio i Svyaz', 1982).

J. Pickands, Upcrossing Probabilities for Stationary Gaussian Process, Transactions of the American Mathematical Society, Vol. 145 (Issue 11):51-73, November 1969.
http://dx.doi.org/10.2307/1995058

V. I. Piterbarg, On the paper by J. Pickands Upcrossing Probabilities for Stationary Gaussian Process, Moscow University Mathematics Bulletin, (Issue 5):25-30, May 1972.
http://dx.doi.org/10.1090/s0002-9947-1969-0250367-x

C. Qualls, and H. Watanabe, Asymptotic Properties of Gaussian Processes, The Annals of Mathematical Statistics, Vol. 43 (Issue 2):580-596, February 1972.
http://dx.doi.org/10.1214/aoms/1177692638

Yu. K. Belyaev, and V. I. Piterbarg, Asymptotics of the Average Number of A-points of Overshoot of a Gaussian-Field Beyond a High Level, Doklady Mathematics, Vol. 13 (Issue 1):309-319, January 1972.

C. Qualls, and H. Watanabe, Asymptotic Properties of Gaussian Random Fields, Transactions of the American Mathematical Society, Vol. 177 (Issue 3):155-171, March 1973.
http://dx.doi.org/10.2307/1996589

V. R. Fatalov, Large Deviations of Gaussian Measures on Spaces lP and LP, p ≥ 2, Theory of Probability and its Applications, Vol. 41 (Issue 3):548-555, March 1996.

V. I. Piterbarg, and V. R. Fatalov, The Laplace method for probability measures in Banach spaces, Russian Mathematical Surveys, Vol. 50 (Issue 6):1151-1239, June 1995.
http://dx.doi.org/10.1070/rm1995v050n06abeh002635

V. I. Piterbarg, and T. L. Mikhaleva, On a Distribution of Maximum of Gaussian Field with a Constant Variance on a Smooth Manifold, Theory of Probability and its Applications, Vol. 41 (Issue 2):367-379, February, 1996.

D. D. Klovskiy, V. A. Soifer, Space-Time Signals Processing [in Russian] (Svyaz’, 1976).

V. L. Levshin, Information Processing in Optical Direction Finding Systems [in Russian] (Mashinostroenie, 1978).

K. K. Vasil’ev, Ya. P. Dragan, V. A. Kazakov, et. al., Applied Theory of Random Processes and Fields [in Russian] (Ulyanovsk State Technical University, 1995).

V. A. Vittikh, V. V. Sergeev, V. A. Soifer, Image Processing in Computer-Aided Research Systems [in Russian] (Nauka, 1982).

P. S. Akimov, P. A. Bakut, V. A. Bogdanovich, et. al., Signal Detection Theory [in Russian] (Radio i Svyaz', 1984).

A. P. Trifonov, Yu. S. Shinakov, Joint Discrimination of Signals and Estimation of Their Parameters Against Background [in Russian] (Radio i Svyaz', 1986).

A. P. Trifonov, E. P. Nechaev, V. I. Parfenov, Detection of Stochastic Signals with Unknown Parameters [in Russian] (Voronezh State University, 1991).

L. A. Shepp, Radon-Nykodym Derivaties of Gaussian Measures, The Annals of Mathematical Statistics, Vol. 37 (Issue 4):321-354, April 1966.
http://dx.doi.org/10.1214/aoms/1177699516


Refbacks

  • There are currently no refbacks.



Please send any question about this web site to info@praiseworthyprize.com
Copyright © 2005-2024 Praise Worthy Prize