On the accuracy of approximation of distributions of sums of independent random variables – which are nonzero with a small probability – by means of accompanying laws

Let $G_i=(1-p_i)E+p_iB_i$ where $0\le p_i\le 1$, $E$ is the distribution concentrated at zero, $B_i$ is an arbitrary one-dimensional distribution, $\displaystyle p=\max_{1\le i\le n}p_i$. Define
$$ G=\prod_{i=1}^nG_i,\qquad D=\prod_{i=1}^n\exp(G_i-E). $$
Then
$$ \sup_x|G\{(-\infty,x)\}-D\{(-\infty,x)\}|\le cp. $$