Precise Laplace-type asymptotics for moderate deviations of the distributions of sums of independent Banach-valued random elements

Formulas are deduced allowing one to find precise asymptotics of moderate deviations for the distributions of sums of independent identically distributed Banach-valued random elements. This result is proved by the Laplace method in Banach spaces. This method is an extension of the classical asymptotic Laplace method to the case of integrals with respect to probability measures in infinite-dimensional Banach spaces. By means of the theorem established in the present paper we find asymptotic representations for the probabilities of moderate deviations of statistics of the form $\omega_n^p$, $p\ge 2$.