  Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$
$\times$ + X n of independent and identically distributed real-valued random
variables have been extensively investigated , in particular when X 1 is not
exponentially integrable. For instance, A.V. Nagaev formulated exact
asymptotics results for P(S n > x n) when X 1 has a semiexponential
distribution (see, [16, 17]). In the same setting, the authors of [4] derived
deviation results at logarithmic scale with shorter proofs relying on classical
tools of large deviation theory and expliciting the rate function at the
transition. In this paper, we exhibit the same asymptotic behaviour for
triangular arrays of semiexponentially distributed random variables, no more
supposed absolutely continuous.
