Let \(\{\) X(t): \(t\geq 0\}\) be a measurable, separable, and continuous in probability stochastic process whose distribution coincides with that of \(\{a^{-k}X(at):\) \(t\geq 0\}\) for some \(a,k>0\). Furthermore, assume that \(P(A)=0\) or 1 whenever \(A\in \sigma (X(t)\), \(t\geq 0)\) and \(P(S_ a^{- 1}A\Delta A)=0\), \(a>0\), where \(S_ aX(t)=a^{-k}X(at)\). Under these hypotheses, the author goes on with his study in Osaka J. Math. 26, 159- 189 (1989). For instance, he proves that \(P(\cup_{s>0}\cap_{t>s}\{| X(t)| \leq t^ kg(t)\})=0\) or 1 whenever g is a positive function which is monotone on some interval (N,\(\infty)\).