Divergence of an integral of a process with small ball estimate

Abstract

The paper contains sufficient conditions on the function $f$ and the stochastic process $X$ that supply the rate of divergence of the integral functional ${\int }_0^{T}f(X_t{)}^{2}dt$ at the rate $T^{1-\epsilon }$ as $T\to \infty$ for every $\epsilon >0$. These conditions include so called small ball estimates which are discussed in detail. Statistical applications are provided.