On Approximations of Distribution Functions of Sums by Infinitely Divisible Laws

Let $\mathfrak{R}(l)$ be a set of distribution functions of random variables $\zeta$ such that $|\zeta|<l,\mathbf D\zeta=1$, $\mathfrak{G}$ a set of infinitely divisible laws and $\xi_1,\xi_2,\dots,\xi_n$ a sequence of independent identically distributed random variables. We put
$$F(x)=\mathbf P\left\{\xi_j<\right\},F^n(x)=\mathbf P\left\{{\xi_1 +\cdots+\xi_n< x}\right\},\\\rho(F,\mathfrak G)=\inf\limits_{G\in\mathfrak G}\sup\limits_x|F(x)-G(x)|$$
and
$$\psi_1(n)=\sup\limits_F\rho(F^n,\mathfrak G),\quad\psi(n,l)=\inf\limits_{F\in\mathfrak N(l)}\rho(F^n,\mathfrak G).$$
Then, for $n\to\infty$
1. $n^{-2/3}(\ln n)^{- 3/2}u(n)=o(\psi _1 (n))$;
2. $n^{- k+1}(\ln n)^{-2k-1/2}u(n)=o(\psi(n,l))\,{\text{when}}\,l < L_{2k}$;
3. $\psi (n,l)u(n)=o(n^{- k} )$ when $l> L_{2k}$,
where $1=L_1= L_2= L_3<L_4=L_5<\cdots(L_k<\infty )$ are absolute constants defined in §1, $u(n)\to0,n\to\infty $ and $x=o(y)$ are equal $\bigl|\frac{x}{y}\bigr|\to0$, $n\to\infty$.
The first equality is an improvement of Prokhorov’s estimate [2]:
$$\psi_1(n)<C_1{(n\ln n)}^{-1}.$$