Convergence rate for the estimation of impulse response function in the space of continuous functions

Rozora, I. V. (2018). Convergence rate for the estimation of impulse response function in the space of continuous functions. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics(3), 30–36. https://doi.org/10.17721/1812-5409.2018/3.4

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 cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the supremum of the estimation error is found that gives a convergence rate of estimator to impulse response function.Key words: impulse response function, linear time-invariant system (LTI), Gaussian process, crosscorrelogram.Pages of the article in the issue: 30 - 36Language of the article: Ukrainian