The aim of this talk is to prove $\ell^p(\mathbb Z^d)$ inequalities with $1 < p < \infty$, for $\lambda$-jumps for discrete Radon transforms. These inequalities are the $r = 2$ endpoints of the $r$-variational estimates due to Mirek, Stein, and Trojan. This is a joint project with E.M. Stein and P. Zorin-Kranich.