Geodesic
Distribution density of the norm of a stable vector
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 105-114

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Let $B$ be a Banach space, $X$ be a stable $B$-valued random vector with exponent $\alpha\in(0,2)$, a $p(\cdot)$, and $p(\cdot)$ 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 exponent $\alpha\in(1,2)$ with unbounded $p$ and prove that if $X$ is a nondegenerate strictly stable vector with exponent $\alpha\in(0,1)$, then $p$ is bounded. 
@article{ZNSL_1987_158_a8,
     author = {M. A. Lifshits},
     title = {Distribution density of the norm of a stable vector},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {105--114},
     publisher = {mathdoc},
     volume = {158},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a8/}
}
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M. A. Lifshits. Distribution density of the norm of a stable vector. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 105-114. http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a8/