Equivalence of some subcritical properties in continuum percolation

We consider the Boolean model on ${\mathbb{R}}^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $\mathrm{\mid }C\mathrm{\mid }$ denotes its volume. Let $\mathrm{\ell }$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(\mathrm{\mid }C\mathrm{\mid })$ is finite if and only if there exists $A,B>0$ such that $\mathrm{\P }(\mathrm{\ell }\ge n)\le A{e}^{-Bn}$ for all $n\ge 1$.