Asymptotic behaviour of sum of exponentials of moving averages

Let S-n = X-1 +...+ X-n be the nth partial sum of an i.i.d. random variables, (X) over bar(i,k) = Si+k-S-i/k, Z(n)(beta) = Sigma(i=0)(n-k) e(beta k (X) over bar i,k), k = [log n/alpha], and alpha and beta be constants applied to a well chosen measure satisfying a large deviation principle, then we precise the asymptotic behavior of Z(n)(beta) which is different whether beta is smaller or larger than some critical value beta(c).