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This paper explores a conditional Gibbs theorem for a random walkinduced byi.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 isproved that when the summands have light tails with some additional regulatityproperty, then the asymptotic conditional distribution of X_{1} can beapproximated in variation norm by the tilted distribution at point a_{n},extending therefore the classical LDP case.

A conditional limit theorem for random walks under extreme deviation