Large deviations in Banach spaces of continuous functions and their applications
This paper examines the large deviation principle for independent and identically distributed copies of (not necessarily smooth) random fields, \(\tau_i\). Working with the Dudley metric, the author establishes deviations for \(u\in(0,\varepsilon_n n^{1/6})\) for a sequence \(\varepsilon_n\) decreasing to 0 such that \(\varepsilon_n n^{1/6}\) increases to infinity. The main theorem states that as \(n\rightarrow\infty\) \[ \lim\frac{P[\sup_{t\in T}|\sum_1^n \tau_i(t)/ \sqrt{n}|>u]} {P[\sup_{t\in T}|G(t)|>u]} =1, \] for \(G\) a suitable Gaussian field. Only an outline of the proof is provided. The author indicates how this result can be used to handle the error in a Monte Carlo approach to numeric integration.
scientific article; zbMATH DE number 1557738