Probabilities of large deviations for sums of independent random variables with a common distribution function from the domain of attraction of an asymmetric stable law

Consider a sequence of independent random variables $\{X_i\}$ with a common distribution function $V(x)$ from the domain of attraction of a stable law with an index $\alpha\in(1,2)$ and suppose that $\mathsf{E}X_1=0$ and
$$ 0<\liminf_{x\to\infty}\frac{1-V(x)}{V(-x)}e^{g(x)}\le\limsup_{x\to\infty}\frac{1-V(x)}{V(-x)}e^{g(x)}<\infty, $$
where the positive function $g(x)$ tends to infinity and
$$ g(x)x^{-\delta} \text{ increases for} x>x_0 \text{ increases for} \delta<1. $$
The paper obtains an asymptotical representation for the probability $\mathsf{P}\{X_1+\dots+X_n>x\}$, which is true uniformly with respect to all positive $x$ for $n$ tending to infinity.
The case $\alpha=2$ was earlier carefully investigated in [10].