An estimation of a convergence rate for the absorption probability in case of a null expectation

Let $\xi_1,\xi_2,\dots$ be a sequence of mutually independent equally distributed random variables with a distribution function $F_\lambda(x)$ depending on a parameter $\lambda$. Let $\mathbf M\xi_1^2=2\lambda^2$ and $\mathbf M\xi_1=0$. Define $n_x$ as the least integer for which $\zeta_n+x\notin(a,b)$, where $\zeta_n=\sum_{i=1}^n\xi_i$ and $(a,b)$ is a finite interval of the real line. Put
$$P_\lambda(x)=\mathbf P\{\zeta_{n_x}+x\ge b\},\quad x\in(a,b),$$
and
$$c_{3\lambda}=\mathbf M|\xi_1|^3.$$
The following assertion is proved: there exists an absolute constant $L$ such that
$$\sup_{a<x<b}\biggl|P_\lambda(x)-\frac{x-a}{b-a}\biggr|<L\frac{c_{3\lambda}}{\lambda^2(b-a)}.$$