Laplace transform estimates and deviation inequalities

We derive deviation inequalities from non-asymptotic bounds of the log-Laplace transform of a function of N random variables. We assume either that these random variables are independent or that they form a Markov chain. We assume also that the partial finite differences of order one and two of the function are suitably bounded, or more generally that they have some exponential moments. The estimates we get are sharp enough to induce a central limit theorem when N goes to infinity and to prove non-asymptotic "almost Gaussian" deviation bounds.