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   -> Volume 5, Issue 5

Preprint: On the wavelet analysis of weak harmonizable processes
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Jun ZHENG and Zhongjie XIE

PostPosted: Wed Dec 04, 2002 9:53 am    
Subject: Preprint: On the wavelet analysis of weak harmonizable processes
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Preprint: On the wavelet analysis of weak harmonizable processes

Abstract: Harmonizable processes (HP) extend the concept of
stationarity (see Loeve: Probability Theory, 1965). Besides wide sense
stationary processes, they also includ many non-stationary processes.
Weak harmonizable processes (WHP) generalized the definition of HP
(see Rozanov: Spectral analysis of abstract functions, Teor. Veroy. i
Primen., 1959) and possess some good new properties. In this paper,
we present the theory of the wavelet decomposition of WHP and
generalize the conclusions of Ping Wah Wong (see: Wavelet
decomposition of harmonizable random processes, IEEE
Trans. Infor.Theory, Vol.39, Jan. 1994). In order to derive the
decomposition of the process, stochastic integral and bi-measure
theory under non-orthonormal measure are necessary. First. the
mathematical preparation on the study of the WHP is given in the first
part of this paper. A sufficient condition for the exchange of
integrations under stochastis and bi-measure, which not necessary to
be orthonormal measure, has been obtained. This result, together with
the dominant convergence theorem of the strict beta-integral and the
stochastic integral representation of the WHP, is then to be used to
establish the wavelet decomposition of the process. Second, from the
weak-harmonizability of the process it has been proved that the
truncated summation of the wavelet decomposition also possesses the
weak harmonizability. Some linear operations on the stochastic wavelet
decompo- sition, such as addition, differentiation and integration are
considered, it is shown that the sum of two WHP is also weak
harmonizable. Giving the spectral measure corresponding to a WHP which
satisfying certain conditions, then the derivative and its wavelet
decomposition is also presented in this paper. In the case of integral
filtering, it is shown that the output process may be obtained from
the wavelet decomposition of the input process. Finally, we show that
for WHP, when the L^2 derivative exists, then the convergence speed of
the mean-squared approximation at the resolution level J, is
exponentially O(1/2^J) decreased.

Zhongjie Xie
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