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.