  Let $X_1,...,X_n$ be $n$ independent unbounded real random variables which
have common, roughly speaking, light-tailed type distribution. Denote by
$S_1^n$ their sum and by $\pi^{a_n}$ the tilted density of $X_1$, where $a_n
\rightarrow\infty$ as $n\rightarrow \infty$. An Abelian type theorem is given,
which is used to approximate the first three centered moments of the
distribution $\pi^{a_n}$. Further, we provide the Edgeworth expansion of
$n$-convolution of the normalized tilted density under the setting of a
triangular array of row-wise independent summands, which is then applied to
obtain one local limit theorem conditioned on extreme deviation event
$(S_1^n/n=a_n)$ with $a_n\rightarrow \infty$.
