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Distribution density of the norm of a stable vector

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Abstract

Let B be a Banach space,X be a stable B -valued random vector with exponentd∈(0,2), and P(·) be the distribution density of the norm of X. In this paper we study the question of the boundedness of P. In particular, we construct examples of a space B with a symmetric stable vector X with exponentd∈(1,2) with unbounded P and prove that if X is a nondegenerate strictly stable vector with exponentd∈(0,1), then P is bounded.

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Literature cited

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Authors
  1. M. A. Lifshits
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 105–114, 1987.

The author is grateful to Yu. A. Davydov, V. I. Paulauskas, V. Yu. Bentkus, and D. Pap for stimulating discussions of the subject of this paper. When the paper was finished the author learned that similar results are found in [9].

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Lifshits, M.A. Distribution density of the norm of a stable vector. J Math Sci 43, 2810–2817 (1988). https://doi.org/10.1007/BF01129895

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  • DOI: https://doi.org/10.1007/BF01129895

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