On the smoothing of probabilities for the sums of integer valued random variables

Let $\xi_1,\xi_2,\dots$ be integer valued independent random variables and $S_n=\xi_1+\dots+\xi_n$. We introduce the function
$$ \sum_m|\mathbf P\{S_n=m\}-\mathbf P\{S_n=m-1\}| $$
as a measure of the «smoothness» of the distribution $S_n$. Some properties of this function are investigated.