Let \(\mathbb{P}\) be a probability measure on \(\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)\) and denote by \(\mathbb{P}^{\star n}\) the \(n\)-fold convolution product \(\mathbb{P} \star \mathbb{P} \star \cdots \star \mathbb{P}\). If \(\int|\omega| \mathbb{P}(d \omega)<\infty\), then

\[\int|\omega| \mathbb{P}^{\star n}(d \omega) \leqslant n \int|\omega| \mathbb{P}(d \omega) \quad \text { and } \quad \int \omega \mathbb{P}^{\star n}(d \omega)=n \int \omega \mathbb{P}(d \omega)\]