On the measure concentration of infinitely divisible distributions

Let $\mathcal{I}$ be the set of all infinitely divisible random variables\ with finite second moments, ${\mathcal{I}}_0=\{X\in \mathcal{I}:\mathrm{V}\mathrm{a}\mathrm{r}(X)>0\}$, $P_{\mathcal{I}}={\inf }_{X\in \mathcal{I}}P\{\mathrm{\mid }X-E[X]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)}\}$ and $P_{{\mathcal{I}}_0}={\inf }_{X\in {\mathcal{I}}_0}P\{\mathrm{\mid }X-E[X]\mathrm{\mid }<\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)}\}$. Firstly, we prove that $P_{\mathcal{I}}\ge P_{{\mathcal{I}}_0}>0$. Secondly, we find the exact values of ${\inf }_{X\in \mathcal{J}}P\{\mathrm{\mid }X-E[X]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)}\}$ and ${\inf }_{X\in \mathcal{J}}P\{\mathrm{\mid }X-E[X]\mathrm{\mid }<\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)}\}$ for the cases that $\mathcal{J}$ is the set of all geometric random variables, symmetric geometric random variables, Poisson random variables and symmetric Poisson random variables, respectively. As a consequence, we obtain that $P_{\mathcal{I}}\le e^{-1}{\sum }_{k=0}^{\mathrm{\infty }}\frac{1}{2^{2k}(k!{)}^2}\approx 0.46576$ and $P_{{\mathcal{I}}_0}\le e^{-1}\approx 0.36788$.