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 \[ {\mathsf P}\left \{\sup _{0 \leq t \leq T} \left | {\varsigma _n (t) -\frac \sigma \gamma n^{ - 1/2} W(\lambda (nt))} \right | >r_n \right \} \leq \alpha _n, \] where $r_n\to 0$ and $\alpha _n \to 0$ as $n \to +\infty$.