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Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$$\times$ + X n of independent and identically distributed real-valued randomvariables have been extensively investigated, in particular when X 1 is notexponentially integrable. For instance, A.V. Nagaev formulated exactasymptotics results for P(S n > x n) when x n > n 1/2 (see, [13, 14]). In thispaper, we derive rough asymptotics results (at logarithmic scale) with shorterproofs relying on classical tools of large deviation theory and expliciting therate function at the transition.

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