Divergence of an integral of a process with small ball estimate

The paper contains sufficient conditions on the function $f$ and the stochastic process $X$ that supply the rate of divergence of the integral functional $\int_0^Tf(X_t)^2dt$ 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.