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A bivariate stable characterization and domains of attraction

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  • Resnick, Sidney
  • Greenwood, Priscilla

Abstract

Bivariate stable distributions are defined as those having a domain of attraction, where vectors are used for normalization. These distributions are identified and their domains of attraction are given in a number of equivalent forms. In one case, marginal convergence implies joint convergence. A bivariate optional stopping property is given. Applications to bivariate random walk are suggested.

Suggested Citation

  • Resnick, Sidney & Greenwood, Priscilla, 1979. "A bivariate stable characterization and domains of attraction," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 206-221, June.
    • Handle: RePEc:eee:jmvana:v:9:y:1979:i:2:p:206-221
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    Cited by:

    1. Jungjun Choi & In Choi, 2019. "Maximum likelihood estimation of autoregressive models with a near unit root and Cauchy errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1121-1142, October.
    2. Chan, Ngai Hang & Zhang, Rong-Mao, 2009. "Quantile inference for near-integrated autoregressive time series under infinite variance and strong dependence," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4124-4148, December.
    3. Cavaliere, Giuseppe & Georgiev, Iliyan, 2013. "Exploiting Infinite Variance Through Dummy Variables In Nonstationary Autoregressions," Econometric Theory, Cambridge University Press, vol. 29(6), pages 1162-1195, December.
    4. Zhiyi Chi, 2018. "On a Multivariate Strong Renewal Theorem," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1235-1272, September.
    5. repec:bot:quadip:118 is not listed on IDEAS
    6. Hasan, Mohammad N., 2001. "Rank tests of unit root hypothesis with infinite variance errors," Journal of Econometrics, Elsevier, vol. 104(1), pages 49-65, August.
    7. Phillips, Peter C.B., 1995. "Robust Nonstationary Regression," Econometric Theory, Cambridge University Press, vol. 11(5), pages 912-951, October.
    8. Kozubowski, Tomasz J. & Meerschaert, Mark M. & Panorska, Anna K. & Scheffler, Hans-Peter, 2005. "Operator geometric stable laws," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 298-323, February.
    9. Wu, Chufang, 1997. "New characterization of Marshall-Olkin-type distributions via bivariate random summation scheme," Statistics & Probability Letters, Elsevier, vol. 34(2), pages 171-178, June.
    10. Meerschaert, Mark M. & Scheffler, Hans-Peter, 1999. "Moment Estimator for Random Vectors with Heavy Tails," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 145-159, October.
    11. Mazur, Stepan & Otryakhin, Dmitry & Podolskij, Mark, 2018. "Estimation of the linear fractional stable motion," Working Papers 2018:3, Örebro University, School of Business.
    12. Buraczewski, Dariusz & Dyszewski, Piotr & Iksanov, Alexander & Marynych, Alexander, 2020. "Random walks in a strongly sparse random environment," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3990-4027.

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