On the closeness of the distributions of the two sums of independent random variables

Let $\{\xi_j\}$, $j=1,2,…,n$ (resp. $\{\eta_j\}$, $j=1,2,…,n$) be independent random variables with distribution functions $\{F_j\}$, $j=1,2,…,n$ (resp. $\{G_j\}$, $j=1,2,…,n$) and let $F$ (resp. $G$) be the distribution function of the sum $\xi=\xi_1+…+\xi_n$ (resp. $\eta=\eta_1+…+\eta_n$).
Let us denote
$$ \mu(k)=\sum_{j=1}^n|\int x^kd(F_j-G_j)|,\quad \nu(r)=\sum_{j=1}^n\int|x|^r|d(F_j-G_j)|. $$
We suppose that $\mu(0)=\mu(1)=…=\mu(m)=0$ and $\nu(r)$ exist for some $r$, $m\le r\le m+1$. In this case
a) if the distribution of $\eta$ has a density bounded by a constant $q$, then
$$ |F(x)-G(x)|<C[\nu(r)q^r]^\frac1{1+r},\eqno{(\text*)} $$

b) if $F$ and $G$ are lattice distributions with the same points of discontinuity and the same largest common factor of the length of the intervals between jumps $h$, then
$$ |F(x)-G(x)|<C_1[\nu(r)h^{-r}]\eqno{(**)} $$
where $C$ and $C_1$ are constants depending only on $m$ and $r$.
In the case a) an estimation of the type (**), which is better then one of the type (*) can be achieved only when some additional requirements on $\xi_j$ are satisfied. The estimations (*) and (**) make it possible to formulate some sufficient conditions for $F$ to converge to infinitely divisible distribution $G$ when the summands $\xi_j$ are not necessarily uniformly infinitesimal.