An Almost Sure Result on Approximation of Homogeneous Random Field from Local Averages

SONG Zhanjie; ZHANG Shuo

doi:10.1049/cje.2018.11.001

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SONG Zhanjie and ZHANG Shuo, “An Almost Sure Result on Approximation of Homogeneous Random Field from Local Averages,” Chinese Journal of Electronics, vol. 28, no. 1, pp. 93-99, 2019, doi: 10.1049/cje.2018.11.001

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SONG Zhanjie and ZHANG Shuo, “An Almost Sure Result on Approximation of Homogeneous Random Field from Local Averages,” Chinese Journal of Electronics, vol. 28, no. 1, pp. 93-99, 2019, doi: 10.1049/cje.2018.11.001

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An Almost Sure Result on Approximation of Homogeneous Random Field from Local Averages

doi: 10.1049/cje.2018.11.001

SONG Zhanjie¹,
ZHANG Shuo^{1,2
,}

1.
School of Mathematics, Tianjin University, Tianjin 300354, China;
2.
Department of Mathematics, Tianjin University of Finance and Economics, Tianjin 300222, China

Funds: This work is supported by the National Natural Science Foundation of China (No.61379014, No.91746107) and the Natural Science Foundation of Tianjin (No.16JCYBJC15900).

More Information

Corresponding author: ZHANG Shuo (corresponding author) was born in Hebei Province, China. She received the M.S. and Ph.D. degrees in Tianjin University. She is a university lecturer of Tianjin University of Finance and Economics. Her research interests include random field and sampling approximation. (Email:shuozhang@tju.edu.cn)
Received Date: 2016-07-25
Rev Recd Date: 2017-05-03
Publish Date: 2019-01-10

Abstract

Abstract

The problem of approximation of homogeneous random field from asymmetric local average sampling is considered in this paper. As a general sampling result, a sufficient condition is obtained to ensure the homogeneous random field be approximated from local averages with probability 1, which extended the result for weak sense stochastic process from local averages to homogeneous random field.
- Sampling theorem,
- Homogeneous random field,
- Local averages sampling

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References

A. V. Balakrishnan, "A note on the sampling principle for continuous signals", IRE Transactions on Information Theory, Vol.3, No.2, pp.143-146, 1957.

Y. K. Belyaev, "Analytic random processes", Theory of Probability and Its Applications, Vol.4, No.4, pp.402-409, 1959.

S. P. Lloyd, "A sampling theorem for stationary (wide sense) stochastic processes", Transactions of the American Mathematical Society, Vol.92, No.1, pp.1-12, 1959.

T. Pogány, "An approach to the sampling theorem for continuous time processes", Australian and New Zealand Journal of Statistics, Vol.31, No.3, pp.427-432, 1989.

M. Zakai, "Band-limited functions and the sampling theorem", Information and Control, Vol.8, No.2, pp.143-158, 1965.

W. Splettstösser, "Sampling series approximation of continuous weak sense stationary processes", Information and Control, Vol.50, No.3, pp.228-241, 1981.

H. Huo and W. Sun, "Sampling theorems and error estimates for random signals in the linear canonical transform domain", Signal Processing, Vol.111, pp.31-38, 2015.

K. Gröchenig, "Reconstruction algorithms in irregular sampling", Mathematics of Computation, Vol.59, No.59, pp.181-194, 1992.

P.L. Butzer and J. Lei, "Errors in truncated sampling series with measured sampled values for non-necessarily bandlimited functions", Funct. Approx. Comment. Math., Vol.26, pp.25-39, 1998.

P. L. Butzer and J. Lei, "Approximation of signals using measured sampled values and error analysis", Communications in Applied Analysis, Vol.4, No.2, pp.245-256, 2000.

W. Sun and X. Zhou, "Reconstruction of band-limited signals from local averages", IEEE Transactions on Information Theory, Vol.48, No.11, pp.2955-2963, 2002.

Z. Song, S. Yang and X. Zhou, "Approximation of signals from local averages", Applied Mathematics Letters, Vol.19, No.12, pp.1414-1420, 2006.

Z. Song, X. Zhou and G. He, "Error estimate on nonbandlimited random signals by local averages", In:V.N. Aleksandrov, et al., eds. ICCS 2006, Part I, Lecture Notes in Computer Sciences 3991. Berlin:Springer-Verlag, pp.822-825, 2006.

Z. Song, W. Sun, S. Yang, et al., "Approximation of weak sense stationary stochastic processes from local averages", Science in China Series A, Vol.50, No.4, pp.457-463, 2007.

G. He and Z. Song, "Approximation of WKS sampling theorem on random signals", Numerical Functional Analysis and Optimization Vol.32, No.32, pp.397-408, 2011.

G. Faÿ and S. Kang, "Average sampling of band-limited stochastic processes", Applied and Computational Harmonic Analysis, Vol.35, No.3, pp.527-534, 2013.

A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers", In:V.M. Tikhomirov, eds., Dokl. Akad. Nauk SSSR, Vol.30, pp.299-303, 1941.

A. M. Yaglom, "Some classes of random fields in n-dimensional space, related to stationary random processes". Theory of Probability and Its Applications, Vol.2, No.3, pp.273-320, 1957.

A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions, Volume I:Basic Results, New York:Springer-Verlag, Vol.131, 1987.

A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions, Volume Ⅱ:Supplementary Notes and References, Berlin:Springer-Velag, 1987.

E. Vanmarcke, Random Fields:Analysis and Synthesis, Singapore:World Scientific, 2010.

V. F. Gaposhkin, "Criteria for the strong law of large numbers for some classes of second-order stationary processes and homogeneous random fields",Theory of Probability and Its Applications, Vol.22, No.2, pp.286-310, 1978.

W. Yang and W. Liu, "Strong law of large numbers and Shannon-McMillan theorem for Markov chain fields on trees", IEEE Transactions on Information Theory, Vol.48, No.1, pp.313-318, 2002.

W. Peng, W. Yang and P. Chen, "Markov approximation of arbitrary random field on homogeneous trees", Journal of Inequalities and Applications, Vol.2012, No.1, pp.1-8, 2012.

S. Zhang, Z. Song and Y. Li, "Approximation of Homogeneous Random Field from local averages", Communications in Statistics-Theory and Methods. Vol.46, No.8, pp.3864-3877, 2017.

G. He and Z. Song, "An almost sure result on sampling of bandlimited random signals", Chinese Journal of Electronics, Vol.22, No.4, pp.747-750, 2013.

W. Splettstösser, R. L. Stens and G. Wilmes, "On the approximation of the interpolating series of G. Valiron", Funct. Approx. Comment. Math., Vol.11, pp.39-56, 1980.

W. Beckner, "Inequalities in Fourier analysis", Annals of Mathematics, Vol.102, No.1, pp.159-182, 1975.

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