A central limit theorem for additive random functions

Let $F(U)$ be an additive real-valued random function defined on bounded Borel subsets $U\subset R^n$ ($U'\bigcap U''=\varnothing$ implies $F(U'\bigcup U'')=F(U')+F(U'')$ with finite variance $\sigma^2(U)$ and $\mathbf MF(U)=0$.
Four types of conditions: А), Б), В) and Г) are studied which guarantee that
$$ \lim_{k\to\infty}\mathbf P\biggl(\frac{F(U_k)}{\sigma(U_k)}<a\biggr)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^a e^{-x^2/2}dx; $$

A) imposes restrictions on the growth of $\sigma^2(U)$ relative to $|U|$,
Б) estimates the absolute moments $C_{2+\delta}(U)=\mathbf M|F(U)|^{2+\delta}$, $\delta>0$,
B) contains various conditions of almost-independence of $F(U')$ and $F(U'')$ if $U'$ and $U''$ are located far from each other,
Г) specifies the meaning of $U_k\to\infty$.
Combinations of such conditions are specified in different theorems. Theorem 1 generalizes the corresponding result of Yu. A. Rozanov [2] even in the case $n=1$ under a milder condition ${\rm B}_1$) . The method of the proof can be traced up to S. N. Bernstein's paper [1]. The results are immediately generalized for functions $F(U)$ on lattice spaces.