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 $\int_0^Tf(X_t)^2dt$ at the rate $T^{1-ε}$ as $T\to\infty$ for every $ε>0$. These conditions include so called small ball estimates which are discussed in detail. Statistical applications are provided.

• The paper establishes explicit divergence rates for integral functionals under small ball probability conditions using analytic and probabilistic methods.
• It employs probabilistic decomposition and lower bound techniques to prove almost sure divergence without relying on ergodicity or stationarity of the process.
• The findings improve statistical estimation in SDEs by ensuring strong consistency, and they extend the analysis to non-Markovian and heavy-tailed processes.

Divergence Rate of Integral Functionals for Processes with Small Ball Estimates

Introduction

This paper investigates the asymptotic divergence rate of integral functionals $\int_0^T f(X_t)^2 \, dt$ for real-valued stochastic processes $X$ and measurable functions $f$ under specific analytic and probabilistic assumptions. While the almost sure (a.s.) divergence or convergence properties of such functionals have been previously established for various subclasses of processes and functions, this work advances the analysis by providing explicit divergence rates under conditions involving "small ball" probabilities. These rates are critical for the strong consistency of statistical estimators in parametric inference for SDEs and for understanding the regularity of additive functionals in ergodic and non-ergodic regimes.

Analytic and Probabilistic Conditions

Conditions on the Integrand Function $f$

A central technical condition, (A1), imposes lower bounds (uniform in space) on the supremum of $|f|$ within sliding intervals, essentially requiring $f$ to grow sufficiently fast (at least polynomially, unless localized nulls are very limited in density). Equivalent forms and sufficient criteria are explored, showing that polynomials and trigonometric sums satisfy (A1), but functions like the exponential may not due to degeneracies near $-\infty$. A higher-order derivative test yielding a lower bound for $|f^{(d)}|$ on all intervals of a fixed length provides a sufficient condition.

Small Ball Estimates for the Process $X$

The main probabilistic component is a "relaxed" small ball estimate (A2, (iii)), bounding the probability that the process $X$ remains in a ball of radius $n$ over an interval of length $\Delta$, uniformly over time intervals. Specifically, for all sufficiently small $n$ and $\Delta$, and some $\lambda,\gamma > 0$, the probability is dominated by $\exp(-K_2 n^{-\lambda} \Delta^{-\gamma})$. Technical forms and their scaling regimes are justified using Gaussian processes with two-sided variational bounds (fractional Brownian motion, subfractional Brownian motion, Ornstein-Uhlenbeck, periodic Brownian bridge), and the relationship to the behaviour of the variance of increments is established.

A stronger, less flexible version (A2, (iv)) is also provided, demanding uniformity over the rectangle of scales $(n,\Delta)$, but the main results focus on the more general relaxed condition, which encompasses broader classes of Markov and non-Markov processes.

Main Theoretical Results

Divergence Theorems

The first principal result (Theorem 3.1) asserts that, under the analytic and small ball conditions stated above, the integral functional diverges a.s. at least at any rate $T^{1-\epsilon}$ as $T\to\infty$ for all $\epsilon>0$:

$$T^{-1+\epsilon}\int_0^T f(X_t)^2\,dt \to \infty \quad \text{a.s.}$$

This is proven through probabilistic decomposition (partitioning time using intervals at scales adapted to small ball probabilities), pointwise lower bounding of $|f(X_t)|$ by mean value arguments, and the application of the Borel–Cantelli lemma using the exponential bounds. The method of proof avoids relying on the ergodicity or stationarity of $X$, making it robust to many non-standard processes.

An extension (Theorem 3.2) replaces moment boundedness of $X$ with a lower polynomial bound on $|f(x)|$ at infinity, capturing cases where $X$ can be unbounded (e.g., Lévy processes).

For self-similar processes $X$ with index $H$ (e.g., fractional Brownian motion), Theorem 3.3 sharpens the result, showing that with $f(x) = |x|^p$, $T^{-1-\epsilon}\int_0^T |X_t|^pdt$ diverges a.s. for all $\epsilon < pH$, with more refined rates available by classical LIL-type results.

Ergodic and Uniformly Integrable Regimes

The ergodic setting is addressed via an application of Birkhoff's individual ergodic theorem, showing that for stationary $X$ and integrable $f(X_t)^2$, the normalized integral converges a.s. to the expectation, confirming the optimal rate of divergence is $T$ when $X$ is sufficiently mixing.

Statistical Applications

The paper applies its divergence results to estimation problems for diffusion and fractional models. For the Ornstein-Uhlenbeck process with unknown drift, strong consistency for the drift estimator is deduced by controlling the rate of divergence of the integrated squared process in the denominator:

$$\hat\theta_T = \frac{\int_0^T Y_t dY_t}{\int_0^T Y_t^2\, dt}$$

This follows by verifying all theoretical assumptions with $f(x) = x$, and employing the strong law for martingales.

An analogous argument applies for the fractional case with drift estimated by regression on the observed process $X_t$ against a function of another process $Y_t$, when $g = f^2$ satisfies the analytic conditions and $Y_t$ meets the small ball requirements.

Implications and Future Directions

This work rigorously establishes that small ball estimates serve as a sufficient framework for ensuring the rapid divergence of integral functionals, even in non-ergodic settings and for non-Markovian or heavy-tailed processes. The results have direct implications for the construction of strongly consistent estimators in parametric inference for SDEs, fBM-driven models, and various "rough" stochastic dynamics. From a theoretical perspective, these results clarify how local regularity (Hölder continuity), tail behaviour, and small ball probabilities interact to determine large time asymptotics of path-integral functionals.

Future work could generalize these criteria to multi-dimensional processes, rough-path integrals, or additive functionals under weaker or path-dependent small ball regimes. Extensions might also address non-polynomial growth functions $f$, more general forms of long-range dependence, or connections with large deviation principles for occupation measures.

Conclusion

The study delivers comprehensive sufficient conditions for the almost sure rate of divergence of integral functionals of the form $\int_0^T f(X_t)^2\, dt$ based on analytic growth and local small ball probability estimates. These results encompass a diverse range of stochastic processes, include constructive examples, and are directly relevant both for asymptotic probability theory and for the statistical estimation of parameters in SDE-type models. The framework significantly expands the toolkit for analyzing the long-time behaviour of additive functionals outside the classical ergodic domain.