Divergence of an integral of a process with small ball estimate

TL;DR

This paper establishes conditions under which the integral of a process's squared function diverges at a specific rate, utilizing small ball estimates, with applications in statistics.

Contribution

It provides new sufficient conditions involving small ball estimates for the divergence rate of integral functionals of stochastic processes.

Findings

The integral diverges at rate T^{1-ε} under specified conditions.

Small ball estimates are crucial for analyzing divergence.

Applications in statistical inference are demonstrated.

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 $\int_0^Tf(X_t)^2dt$ 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.