Tail exactness of multivariate saddlepoint approximations

We consider a log-concave density fin R-m satisfying certain weak condition s, particularly on the Hessian matrix of log f. For such a density, we prov e tail exactness of the multivariate saddlepoint approximation. The proof i s based on a local limit theorem for the exponential family generated by f. However, the result refers not to asymptotic behaviour under repeated samp ling, but to a limiting property at the boundary of the domain of f. Our ap proach does not apply any complex analysis, but relies totally on convex an alysis and exponential models arguments.