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On the product of two harmonizable time series

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  • Dehay, Dominique

Abstract

In order to state sufficient conditions for the harmonizability of the product time series of two harmonizable time series, the notion of Lp-harmonizable time series is introduced for 1 [less-than-or-equals, slant] p [less-than-or-equals, slant] + [infinity]. Then, the problem of the product of two stochastic measures is tackled and Fubini type theorems are deduced. We derive sufficient conditions for the harmonizability of a weighted convolution time series of two harmonizable time series. As an application to nonlinear prediction theory, asymptotically unbiased estimors for values of the cross spectral bimeasure of two harmonizable time series are given. The L1-convergence of these estimators towards some random variables is established from the law of large numbers stated for Lp-harmonizable series. Sufficient conditions for the a.e. convergence are obtained from the strong law of large numbers. The case of two jointly stationary harmonizable series is also considered. The results apply to the estimation of the asymptotic spectral measure of some asymptotically stationary series.

Suggested Citation

  • Dehay, Dominique, 1991. "On the product of two harmonizable time series," Stochastic Processes and their Applications, Elsevier, vol. 38(2), pages 347-358, August.
    • Handle: RePEc:eee:spapps:v:38:y:1991:i:2:p:347-358
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    Cited by:

    1. Jason Hong Jae Park, 2019. "Random Measure Algebras Under O-dot Product and Morse-Transue Integral Convolution," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 8(6), pages 1-73, November.
    2. Boudou, Alain & Romain, Yves, 2010. "On the integral with respect to the tensor product of two random measures," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 385-394, February.

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