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I am looking for an estimate of the following sum/expectation: \begin{align*}%$ J_n & = \mathbb{E}\left( e^{n f(X_n) + \log(n) g(X_n) + h(X_n)} \right) \\ & = \frac{1}{2^n} \sum_{k = 0}^n {n \choose k} e^{n f(2k/n - 1)}\times n^{ g(2k/n - 1) } \times e^{ h(2k/n - 1)} \end{align*} where $ f, g, h $ are bounded continuous/$ C^\infty $ functions (say), and $ X_n = S_n/n $ with $ S_n $ a simple random walk (increments $ \pm 1 $ with probability $ \frac{1}{2} $), namely $ S_n = 2 \mathrm{Bin}(n, \frac{1}{2}) - n $ in law.

The Varadhan lemma [1] gives (under some assumptions) \begin{align*}%$ \frac{1}{n} \log \mathbb{E}\left( e^{n f(X_n) } \right) \underset{n \rightarrow +\infty }{\longrightarrow} \sup_{(-1, 1)}\{ f - I \} \end{align*} with \begin{align*}%$ I(x) = \frac{1+ x}{2}\log(1 + x) + \frac{1 - x}{2} \log(1 - x) \end{align*} and the Bahadur-Rao theorem [2] gives for $ x > \frac{1}{2} $ \begin{align*}%$ \mathbb{P}\left( X_n > x \right) = \frac{ \eta_x^2 \sqrt{\Lambda''(\eta_x) } }{1 - e^{- \eta_x } } \, \sqrt{2\pi n} \, e^{n \Lambda^*(x) }(1 + o_x(1) ) \end{align*} with \begin{align*}%$ \Lambda(s) & = \log \mathbb{E}\left( e^{ s X_1 } \right) \\ \Lambda'(\eta_x) & = x \\ \Lambda^*(x) & = \sup\{ x y - \Lambda(y) \} \quad\mbox{(Legendre transform of $ \Lambda $}) \end{align*}

The question I am asking is an improvement of the Laplace version of the Bahadur-Rao theorem. Equivalently, this is the (precise) Laplace method for a Binomial sum.

Has anyone some references on that? I suppose this must be very classical, but I did not find it in the book of Dembo and Zeitouni (the Bahadur-Rao theorem is 3.7.4 in this book).

[1] https://en.wikipedia.org/wiki/Varadhan's_lemma

[2] https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-31/issue-4/On-Deviations-of-the-Sample-Mean/10.1214/aoms/1177705674.full

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