Moderate deviation probabilities for open convex sets: nonlogarithmic behavior. (Q1879825)

Let \(X,X_1,X_2,\ldots\) be independent, identically distributed random vectors where \({\mathcal L}(X)=\mu\), and \(\mu\) is a Borel probability measure on the real separable Banach space \(B\). Let be \(S_n=\sum_{j=1}^nX_j\) and assume \({\mathcal L}(S_n/n^{1/2})\) converges weakly. Let \(\{b_n\}\) be a positive sequence such that \[ b_n/n^{1/2}\to\infty \quad {\text{and}} \quad b_n/n\to0. \tag \(*\) \] Let us consider the sequence of the probabilities \(\{P(S_n/b_n)\in A\}\). These probabilities are called moderate deviation probabilities. The asymptotic behavior of the probabilities under the condition \((*)\) for open convex sets in both the finite- and infinite-dimensional settings are studied. The obtained results are based on the existence of dominating points for these sets, a related representation formula and asymptotics for the integral term of the formula.