Correction and addendum to “On mixtures of the normal distribution by the generalized gamma convolutions”. (English) Zbl 0711.60017
The author revised the corollary in his paper [ibid. 21, 29-41 (1989; Zbl 0691.60014)], and showed the following example of probability distributions which shows the difference between the one-dimensional case and the three-dimensional case. Let us consider the self-decomposability of probability distributions with density
\[
f(x)=((2\pi)^{(d- 1)/2}\Gamma ((d+1)/2)\sqrt{2+| \beta |^ 2})^{-1}e^{- \sqrt{2+| \beta |^ 2}| x| +\beta x},\quad (\beta,x\in R^ d).
\]
Then, when \(d=1\), this distribution is self-decomposable for all \(\beta\in R\), but when \(d=3\) this distribution is self-decomposable only if \(\beta\) is the zero vector.
Reviewer: K.Takano
MSC:
60E07 | Infinitely divisible distributions; stable distributions |
60E10 | Characteristic functions; other transforms |