A conditional limit theorem for random walks under extreme deviation

Abstract: This paper explores a conditional Gibbs theorem for a random walkinduced by i.i.d. (X_{1},..,X_{n}) conditioned on an extreme deviation of its sum (S_{1}^{n}=na_{n}) or (S_{1}^{n}>na_{n}) where a_{n}\rightarrow\infty. It is proved that when the summands have light tails with some additional regulatity property, then the asymptotic conditional distribution of X_{1} can be approximated in variation norm by the tilted distribution at point a_{n}, extending therefore the classical LDP case.