Abstract
Conditional moments E[X 2|p|X 1 = x] of an α-stable random vector (X 1,X 2) may exist even if p ⩾ α. The precise conditions are stated in Cioczek-Georges and Taqqu [4]_and Samorodnitsky and Taqqu [12]. This paper provides the proof for the most delicate cases, namely 1 < α< 2 and p < 2α+ 1, which is the maximal range of possible p’s when the vector (X 1, X 2) is nondegenerate.
This research was supported by the ONR grant N00014-90-J-1287 and NSF grants DMS-9404093 and NCR-9404931 at Boston University.
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References
S. Cambanis and S. B. Fotopoulos. Conditional variance for stable random vectors. Probability and Mathematical Statistics, 15:195–214, 1995.
S. Cambanis, S. B. Fotopoulos, and L. He. On the conditional variance for scale mixtures of normal distributions. Preprint, 1996.
S. Cambanis and W. Wu. Multiple regression on stable vectors. Journal of Multivariate Analysis, 41:243–272, 1992.
R. Cioczek-Georges and M. S. Taqqu. How do conditional moments of stable vectors depend on the spectral measure? Stochastic Processes and their Applications, 54:95–111, 1994.
R. Cioczek-Georges and M. S. Taqqu. Form of the conditional variance for symmetric stable random variables. Statistica Sinica, 5:351–361, 1995.
R. Cioczek-Georges and M. S. Taqqu. Necessary conditions for the existence of conditional moments of stable random variables. Stochastic Processes and their Applications, 56:233–246, 1995.
S. B. Fotopoulos and L. He. Form of the conditional variance-covariance matrix for α-stable scale mixtures of normal distributions. Preprint, 1996.
C. D. Hardin Jr., G. Samorodnitsky, and M. S. Taqqu. Non-linear regression of stable random variables. The Annals of Applied Probability, 1:582–612, 1991.
B. Ramachandran. On characteristic functions and moments. Sankhya, 31(Series A):1–12, 1969.
H. L. Royden. Real Analysis. Macmillan, third edition, 1988.
G. Samorodnitsky and M. S. Taqqu. Conditional moments and linear regression for stable random variables. Stochastic Processes and their Applications, 39:183–199, 1991.
G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York, London, 1994.
W. Wu and S. Cambanis. Conditional variance of symmetric stable variables. In S. Cambanis, G. Samorodnitsky, and M. S. Taqqu, editors, Stable Processes and Related Topics, volume 25 of Progress in Probability, pp. 85–99, Birkhäuser, Boston, 199
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Cioczek-Georges, R., Taqqu, M.S. (1998). Sufficient Conditions for the Existence of Conditional Moments of Stable Random Variables. In: Karatzas, I., Rajput, B.S., Taqqu, M.S. (eds) Stochastic Processes and Related Topics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2030-5_4
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DOI: https://doi.org/10.1007/978-1-4612-2030-5_4
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