On probabilities of moderate deviations

Let $\{X_n;n=1,2,\dots\}$ be a sequence of independent identically distributed random variables, and let $\sigma>0$ and $c>0$. Put
$$ S_n=\sum_{i=1}^n X_i,\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt. $$
The rate of convergence of probabilities $P(S_n\geq\varepsilon(n\log n)^{1/r})$, $P(\max_{1\leq k\leq n}S_k\geq\varepsilon(n\log n)^{1/r})$ and $P(\sup_{k\geq n}\frac{S_k}{(k\log k)^{1/r}}\geq\varepsilon)$ for all $\varepsilon>\varepsilon_0$ and some $r$, $\varepsilon_0$ is studied and necessary and sufficient conditions are found for the relation
$$ P(S_n\geq x\sigma\sqrt n)=(1-\Phi(x))(1+O(1)),\quad n\to\infty,\quad0\leq x\leq C\sqrt{\log n}, $$
to hold.