Let (Y,\(\rho)\) be a complete, separable metric space and \(D=D_{[0,\infty [}(Y)\) be the space of cadlag Y-valued functions on [0,\(\infty [\). The author says that the sequence of processes \((X^ n)_{n\in {\mathbb{N}}}\), with sample paths in D, has tight majorization of jumps (t.m.j.) if for every \(\epsilon >0\) there is a process \(\gamma^{\epsilon}\), such that its sample paths majorizes jumps and \[ P(\rho (X^ n_{t-},X^ n_ t)\leq \gamma^{\epsilon},\text{ for every } t\in {\mathbb{R}}_+)\geq 1-\epsilon \] for any \(n\in {\mathbb{N}}\). Via this notion the relation of tightness of \((X^ n)_{n\in {\mathbb{N}}}\) in \(J_ 1\)-topology and UC-topology on D is described. Some sufficient conditions for the property of t.m.j. are formulated.