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On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields *

Florin Avram, Nikolai Leonenko, Ludmila Sakhno (2010)

ESAIM: Probability and Statistics

Many statistical applications require establishing central limit theorems for sums/integrals S T ( h ) = ∫ t ∈ I T h ( X t ) d t or for quadratic forms Q T ( h ) = ∫ t , s ∈ I T b ^ ( t - s ) h ( X t , X s ) d s d t , where Xt is a stationary process. A particularly important case is that of Appell polynomials h(Xt) = Pm(Xt), h(Xt,Xs) = Pm,n (Xt,Xs), since the “Appell expansion rank" determines typically the type of central limit theorem satisfied by the functionals ST(h), QT(h). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu,...

On discrete Fourier analysis of amplitude and phase modulated signals

Waldemar Popiński (2012)

Applicationes Mathematicae

In this work the problem of characterization of the Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal o t , t=0,1,...,n-1, modulated by random fluctuations of its amplitude and/or phase is investigated. It is assumed that the amplitude and/or phase of the signal at discrete times of observation are distorted by realizations of uncorrelated random variables or randomly permuted sequences of complex numbers. We derive the expected values and bounds on the variances of such...

On discrete Fourier spectrum of a harmonic with random frequency modulation

Waldemar Popiński (2013)

Applicationes Mathematicae

Asymptotic properties of the Discrete Fourier Transform spectrum of a complex monochromatic oscillation with frequency randomly distorted at the observation times t=0,1,..., n-1 by a series of independent and identically distributed fluctuations is investigated. It is proved that the second moments of the spectrum at the discrete Fourier frequencies converge uniformly to zero as n → ∞ for certain frequency fluctuation distributions. The observed effect occurs even for frequency fluctuations with...

On the estimation of the autocorrelation function

Manuel Duarte Ortigueira (2010)

Discussiones Mathematicae Probability and Statistics

The autocorrelation function has a very important role in several application areas involving stochastic processes. In fact, it assumes the theoretical base for Spectral analysis, ARMA (and generalizations) modeling, detection, etc. However and as it is well known, the results obtained with the more current estimates of the autocorrelation function (biased or not) are frequently bad, even when we have access to a large number of points. On the other hand, in some applications, we need to perform...

Optimal random sampling for spectrum estimation in DASP applications

Andrzej Tarczynski, Dongdong Qu (2005)

International Journal of Applied Mathematics and Computer Science

In this paper we analyse a class of DASP (Digital Alias-free Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The estimators are unbiased inside arbitrarily wide frequency ranges, regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, we calculate...

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