Probabilistic proofs of large deviation results for sums of semiexponential random variables and explicit rate function at the transition

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 n > n 1/2 (see, [13, 14]). In this paper, we derive rough asymptotics results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition.