# Sample continuity with probability one for the estimator of impulse response function

## DOI:

https://doi.org/10.17721/1812-5409.2020/3.10## Abstract

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output crosscorrelogram is taken as an estimator of the response function. The conditions on sample continuousness with probability one for impulse response function are investigated.

* Key words*: impulse response function, linear time-invariant system (LTI), Gaussian process, crosscorrelogram, sample continuity.

* Pages of the article in the issue*: 96 - 102

** Language of the article**: Ukrainian

## References

BULDYGIN V., BLAZHIEVSKA I. (2010) Asymptotic properties of cross-correlogram estimators of impulse response functions in linear system. Research Bulletin of National Technical University of Ukraine "KPI", 4, 16–27.

BULDYGIN V., FU LI (1997) On asymptotic normality of an estimation of unit impulse responses of linear system I, II. Theor. Probability and Math. Statist., 54, 55, 3–17, 30–37.

BULDYGIN V., UTZET F., ZAIATS V. (2004) Asymptotic normality of crosscoorrelogram estimators of the response function. Statistical Infernce for Stochastic Processes, 7, 1–34.

KOZACHENKO YU., ROZORA I. (2015) On cross-correlogram estimators of impulse response functions. Theor. Probability and Math. Statist., 93, 75-86.

ROZORA I. (2018) Convergence rate for the estimation of impulse response function in the space of continious functions. Bulletin of KNU. Series: Physics and Mathematics, 3, 30-36.

ROZORA I. (2018) On the convergence rate for the estimation of impulse response function in the space L_p(T). Bulletin of KNU. Series: Physics and Mathematics, 4, 36-41.

KOZACHENKO YU., ROZORA I. (2019) Conditions of sample continuity with probability one for Square-Gaussian stochastic processes. Theor. Probability and Math. Statist., 101, 134-146.

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (3), 96–102. https://doi.org/10.17721/1812-5409.2020/3.10

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