  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.
