Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. II

“Narrow” Zones of Local and Integral Normal Attraction. Using the notation in Part I of this article, we consider the integral normal attraction zones for the variables $X_i$ and local normal attraction zones for $X_j\in(d)$. The monotone function $h(x)\leq x^{1/2}$ is considered under the supplementary conditions explained in Part I; the “narrow zone theorems” are more conveniently expressed in terms of the condition
\begin{equation} \label{eq*}\tag{*} E\exp h(|X_j |)<\infty. \end{equation}
The equation
$$ h(\sqrt n\Lambda(n))=(\Lambda(n))^2 $$
determines the monotone function $\Lambda (n)$. The condition \eqref{eq*} is necessary for the zones $[0,\Lambda (n)\rho (n)],[ - \Lambda (n)\rho (n),0]$ to be z.n.a., and for $X_j \in (d)$ to be z.u.l.n.a. It is sufficientt for the zones $[0,\Lambda (n)/\rho(n)], [-\Lambda(n)/\rho (n),0]$ to be z.n.a. and for $X_j\in(d)$ – to be z.u.l.n.a.