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Sharp Gaussian Approximation Bounds for Linear Systems with α-stable Noise

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Kontoyiannis, Ioannis  ORCID logo
Riabiz, Marina 
Ardeshiri, Tohid 
Godsill, Simon 


We report the results of several theoretical studies into the convergence rate for certain random series representations of α-stable random variables, which are motivated by and find application in modelling heavy-tailed noise in time series analysis, inference, and stochastic processes. The use of α-stable noise distributions generally leads to analytically intractable inference problems. The particular version of the Poisson series representation invoked here implies that the resulting distributions are “conditionally Gaussian,” for which inference is relatively straightforward, although an infinite series is still involved. Our approach is to approximate the residual (or “tail”) part of the series from some point, c > 0, say, to ∞, as a Gaussian random variable. Empirically, this approximation has been found to be very accurate for large c. We study the rate of convergence, as c → ∞, of this Gaussian approximation. This allows the selection of appropriate truncation parameters, so that a desired level of accuracy for the approximate model can be achieved. Explicit, nonasymptotic bounds are obtained for the Kolmogorov distance between the relevant distribution functions, through the application of probability-theoretic tools. The theoretical results obtained are found to be in very close agreement with numerical results obtained in earlier work.



40 Engineering, 4901 Applied Mathematics, 4006 Communications Engineering, 49 Mathematical Sciences, 4905 Statistics

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2018 IEEE International Symposium on Information Theory (ISIT

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