The invariance principle for the Ornstein-Uhlenbeck process with fast Poisson time: An estimate for the rate of convergence

We consider the invariance principle for

$$\varsigma _n (t) = n^{ - 1/2} \int_0^{Z(nt)} \xi (s)\,ds,$$

where $\xi (s)$ is the Ornstein-Uhlenbeck process and $Z(t)$, $t \geq 0$, is the Poisson process such that ${\mathsf E} Z(t) = \lambda (t)$. We prove that

$${\mathsf P}\left\{\sup_{0 \leq t \leq T} \left\vert {\varsigma _n... ...ma\gamma n^{ - 1/2} W(\lambda (nt))} \right\vert >r_n \right\} \leq \alpha _n,$$

where $r_n\to 0$ and $\alpha _n \to 0$ as $n \to+\infty$.