Let $F(x)$ be a distribution function with $F(0)=0$ and $$ H(x,A)=\sum_{k=0}^\infty A^kF_k(x),\quad A>0, $$ where $F_k(x)$, $k\ge1$, is the $k$-th convolution of $F(x)$ and $F_0(x)$ is the degenerate distribution function with the unit jump at $x=0$. In the paper the asymptotic behaviour of $H(x,A)-H(x-l,A)$ is studied. The dominant term and an estimate for the remainder are obtained, $F(x)$ being assumed1) to be of the lattice type or to have a non-zero absolutely continuous component and2) to have a finite number of moments or to satisty well-known Cramér's condition.