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Multiple stochastic integrals with dependent integrators

Author

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  • Fox, Robert
  • Taqqu, Murad S.

Abstract

Let [mu] be a [sigma]-finite measure, R = (rij) be a covariance matrix, and B1,..., Bn be dependent Gaussian measures satisfying EBi(A1) Bj(A2) = rij[mu](A1 [down curve] A2). Multiple integrals of the form In(f) = [integral operator]f(x1,..., xn) dB1(x1) ... dBn(xn), with f [set membership, variant] L2([mu]n) are investigated. A diagram formula is established and a class of functions which play the role of the Hermite polynomials for these more general integrals is introduced. Cumulants of double integrals are evaluated and the following result is established: if {Xj} and {Yj} are correlated stationary sequences of strongly dependent Gaussian random variables, then [Sigma]j=1[Nt] XjYj, adequately normalized, converges in D[0, 1] to I2(fi).

Suggested Citation

  • Fox, Robert & Taqqu, Murad S., 1987. "Multiple stochastic integrals with dependent integrators," Journal of Multivariate Analysis, Elsevier, vol. 21(1), pages 105-127, February.
    • Handle: RePEc:eee:jmvana:v:21:y:1987:i:1:p:105-127
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    Citations

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    Cited by:

    1. Anh, V. V. & Leonenko, N. N., 1999. "Non-Gaussian scenarios for the heat equation with singular initial conditions," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 91-114, November.
    2. Nikolai Leonenko & Ludmila Sakhno, 2001. "On the Kaplan–Meier Estimator of Long-Range Dependent Sequences," Statistical Inference for Stochastic Processes, Springer, vol. 4(1), pages 17-40, January.
    3. Bardet, Jean-Marc & Tudor, Ciprian, 2014. "Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 1-16.
    4. Radomyra Shevchenko & Ciprian A. Tudor, 2020. "Parameter estimation for the Rosenblatt Ornstein–Uhlenbeck process with periodic mean," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 227-247, April.
    5. Noreddine, Salim & Nourdin, Ivan, 2011. "On the Gaussian approximation of vector-valued multiple integrals," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1008-1017, July.
    6. Nikolai Leonenko & Ludmila Sakhno & Emanuele Taufer, 2000. "On the product limit estimator for long range dependent sequences under chi-square subordination," Quaderni DISA 041, Department of Computer and Management Sciences, University of Trento, Italy, revised 12 Sep 2003.
    7. Valery Buldygin & Frederic Utzet & Vladimir Zaiats, 2004. "Asymptotic Normality of Cross-correlogram Estimates of the Response Function," Statistical Inference for Stochastic Processes, Springer, vol. 7(1), pages 1-34, March.
    8. Jin, Hao & Zhang, Jinsuo & Zhang, Si & Yu, Cong, 2013. "The spurious regression of AR(p) infinite-variance sequence in the presence of structural breaks," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 25-40.
    9. Tsay, Wen-Jen & Chung, Ching-Fan, 2000. "The spurious regression of fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 96(1), pages 155-182, May.
    10. Taufer, Emanuele, 2015. "On the empirical process of strongly dependent stable random variables: asymptotic properties, simulation and applications," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 262-271.
    11. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.

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