Moderate deviations for \(m\)-dependent random variables with Banach space values (Q1373981)

A theorem on the moderate deviations for \(m\)-dependent separable Banach space valued random elements is established. Recall that a sequence \((X_n)_{n\geq 1}\) of strictly stationary random elements is called \(m\)-dependent, where \(m\) is a nonnegative integer, if for any \(k\geq 1\), the two collections \(\{X_1,\cdot , X_k\}\) and \(\{X_{k+m+1}, X_{k+m+2},\cdot \}\) are independent. It is proved that if \(\{X_n\}\) is a sequence of \(m\)-dependent random elements and if \(\{b_n\}\) is a sequence of positive numbers such that \(b_n/\sqrt{n} \to \infty\) and \(b_n/n\to 0\) under some assumptions, for a given Borel subset \(A\) and for a certain real rate function \(I\) on the Banach space, for the asymptotic behavior we have \[ P\Biggl\{ \frac{\sum _{k=1}^n X_k}{b_n}\in A\Biggr\}\simeq \exp\Biggl\{-\frac{b_n^2}{n} \inf_{x\in A} I(x)\Biggr\}. \]