Asymptotics deviation probabilities of the sum $S_n=X_1+\dots+X_n$ of independent and identically distributed real-valued random variables have been extensively investigated, in particular when $X_1$ is not exponentially integrable. For instance, Nagaev (1969a, 1969b) formulated exact asymptotics results for $\mathbb{P}(S_n>x_n)$ with $x_n\to \infty$ when $X_1$ has a semiexponential distribution. In the same setting, Brosset et al. (2020) derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large-deviation theory and making the rate function at the transition explicit. In this paper we exhibit the same asymptotic behavior for triangular arrays of semiexponentially distributed random variables.

As an extension of a central limit theorem established by Svante Janson, we prove a Berry–Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.

Many mathematical models involve input parameters, which are not precisely known. Globalsensitivity analysis aims to identify the parameters whose uncertainty has the largestimpact on the variability of a quantity of interest (output of the model). One of thestatistical tools used to quantify the influence of each input variable on the output isthe Sobol sensitivity index. We consider the statistical estimation of this index from afinite sample of model outputs: we present two estimators and state a central limittheorem for each. We show that one of these estimators has an optimal asymptotic variance.We also generalize our results to the case where the true output is not observable, and isreplaced by a noisy version.

In this paper, we consider a new framework where two types of data are available:experimental dataY1,...,Ynsupposed to be i.i.d from Y and outputs from a simulated reduced model.We develop a procedure for parameter estimation to characterize a feature of thephenomenon Y. We prove a risk bound qualifying the proposed procedure interms of the number of experimental data n, reduced model complexity andcomputing budget m. The method we present is general enough to cover awide range of applications. To illustrate our procedure we provide a numericalexample.