Saddle point approximations to probabilities of sample mean deviations (Q1366390)

Let \(X_n\), \(n\in\mathbb N\), be independent, possibly non-identically distributed random variables and define \(S_n=X_1+\dots+X_n\). Assume that all exponential moments \(E[\exp(tX_n)]\) with \(t>0\) and \(n\in\mathbb N\) are finite. Given \(a\in\mathbb R\) satisfying \(na>E[S_n]\) and \(P(S_n>na)>0\) for all \(n\in\mathbb N\), the author proves asymptotic approximations to the probability \(P(S_n\geq na)\) of a sample mean deviation. Let \(\tau_n\) be the argument minimizing \(t\mapsto E[\exp(t(S_n-na))]\) in \((0,\infty)\), and let \(\rho_n\) denote the minimal value. For \(j,n\in\mathbb N\), \(j\leq n\), consider random variables \(U_n\) and \(Y_{n,j}\) conjugate to \(S_n-na\) and \(X_j-a\), respectively, in the sense that their distributions are exponentially tilted using the densities \(\exp(\tau_n(S_n-na))/\rho_n\) and \(\exp(\tau_n(X_j-a))/r_{n,j}\) with \(r_{n,j}=E[\exp(\tau_n(X_j-a))]\). Assuming a so-called Petrov-type condition involving the characteristic functions of the \(Y_{n,j}\), the approximations of \(P(S_n\geq na)\) are derived in terms of \(\rho_n\), \(\tau_n\), the variance of \(U_n\), and the Lyapunov fractions of the \(Y_{n,j}\).