Geodesic
On asymptotic behavior of the error term in cross-correlogram estimation of response functions in linear systems
Teoriâ slučajnyh processov, Tome 16 (2010) no. 2, pp. 5-11.

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of estimation of an unknown response function of a linear system with inner noises is considered. We suppose that the response function of the system belongs to $L_{2}({\mathbb R})$. Integral-type sample input-output cross-correlograms are taken as estimators of the response function. The inputs are supposed to be zero-mean stationary Gaussian processes that are close, in some sense, to a white noise. Both the asymptotic normality of finite-dimensional distributions of the normalized error term in the cross-correlogram estimation and the asymptotic normality in the space of continuous functions are discussed.
Keywords: Response function, sample cross-correlogram, asymptotic normality. 
@article{THSP_2010_16_2_a1,
     author = {I. P. Blazhievska},
     title = {On asymptotic behavior of the error term in cross-correlogram estimation of response functions in linear systems},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {5--11},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a1/}
}
TY  - JOUR
AU  - I. P. Blazhievska
TI  - On asymptotic behavior of the error term in cross-correlogram estimation of response functions in linear systems
JO  - Teoriâ slučajnyh processov
PY  - 2010
SP  - 5
EP  - 11
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a1/
LA  - en
ID  - THSP_2010_16_2_a1
ER  - 
%0 Journal Article
%A I. P. Blazhievska
%T On asymptotic behavior of the error term in cross-correlogram estimation of response functions in linear systems
%J Teoriâ slučajnyh processov
%D 2010
%P 5-11
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a1/
%G en
%F THSP_2010_16_2_a1
I. P. Blazhievska. On asymptotic behavior of the error term in cross-correlogram estimation of response functions in linear systems. Teoriâ slučajnyh processov, Tome 16 (2010) no. 2, pp. 5-11. http://geodesic.mathdoc.fr/item/THSP_2010_16_2_a1/

[1] H. Akaike, “On statistical estimation of the frequency response function of a system having multiple input”, Ann. Inst. Statist. Math., 17 (1965), 185–210

[2] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968

[3] J. S. Bendat, A. G. Piersol, Engineering Applications of Correlation and Spectral Analysis, Wiley, New York, 1993

[4] R. Bentkus, “On asymptotic normality of the estimator of the spectral function”, Lithuanian Math. J., 12:3 (1972), 3–17

[5] D. R. Brillinger, Time Series: Data Analysis and Theory, Holden Day, San Francisco, 1981

[6] V. V. Buldygin, “On the properties of an empirical correlogram of a Gaussian process with square integrable spectral density”, Ukrain. Math. J., 47:7 (1995), 1006–1021

[7] V. V. Buldygin, I. P. Blazhievska, “On asymptotic behavior of cross-correlogram estimators of response functions in linear Volterra systems”, Theory of Stochastic Processes, 15 (31):2 (2009), 84–98

[8] V. V. Buldygin, Fu Li, “On asymptotical normality of an estimation of unit impulse responses of linear systems. I; II”, Theor. Probab. and Math. Statist., 54 (1997), 17–24; 55, 29–36

[9] V. V. Buldygin, Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, American Mathematical Society, Providence, RI, 2000

[10] V. V. Buldygin, V. G. Kurotschka, “On cross-correlogram estimators of the response function in continuous linear systems from discrete observations”, Random Oper. and Stoch. Eq., 7:1 (1999), 71–90

[11] V. Buldygin, F. Utzet, V. Zaiats, “Cross-correlogram estimates of the response function in linear and bilinear Volterra systems”, Prague Stochastics'98, v. 1, eds. Huskova, M., et al., Union of Czech Mathematicians and Physicists, Prague, 1998, 61–66

[12] V. Buldygin, F. Utzet, V. Zaiats, “Asymptotic normality of cross-correlogram estimates of the response function”, Statistical Interference for Stochastic Processes, 7 (2004), 1–34

[13] G. L. Cariolaro, G. B. Di Masi, “Second-order analysis of the input of a discrete-time Volterra system driven by white noise”, IEEE Trans. Inform. Theory, 26 (1980), 175–184

[14] R. M. Dudley, “Sample functions of the Gaussian process”, Ann. of Probab., 1 (1973), 66–103