Large deviations for symmetrised empirical measures

In this paper we prove a Large Deviation Principle for the sequence of symmetrised empirical measures 1/n Sigma(n)(i=1) delta(X-i(n), X-sigma n(n)(i)) where sigma(n) is a random permutation and ((X-i(n)) 1 <= i <= n) n >= 1 is a triangular array of random variables with suitable properties. As an application we show how this result allows to improve the Large Deviation Principles for symmetrised initial-terminal conditions bridge processes recently established by Adams, Dorlas and Konig.