## The invariance principle for the Ornstein-Uhlenbeck process with fast Poisson time: an estimate for the rate of convergence.(Ukrainian, English)Zbl 1199.60099

Teor. Jmovirn. Mat. Stat. 76, 14-22 (2007); translation in Theory Probab. Math. Stat. 76, 15-22 (2008).
The authors study the invariance principle for normalized integrals with fast Poisson time. The problem considered is estimating the rate of approximation in probability of the process $\zeta_n(t)=\frac{1}{\sqrt{n}}\int_0^{Z(nt)}\xi(s)ds, t\in[0,T],$ by the family of processes $$\frac{\sigma}{\gamma}\frac{1}{\sqrt{n}}W(\lambda(nt))$$ as $$n\to+\infty$$. Here $$\xi(t)$$ is the Ornstein-Uhlenbeck process $\xi(t)=\xi(0)-\gamma\int_0^t\xi(s)ds+\sigma W(t),$ where $$\gamma>0$$ and $$\sigma>0$$ are constants and $$W(t),t\geq0,$$ is a standard Wiener process, $$Z(t),t\geq0,$$ is a Poisson process such that $$EZ(t)=\lambda(t),\lambda(0)=0$$. It is proved that the following inequality holds true $P\left\{ \sup_{0\leq t\leq T}\left|\zeta_n(t)-\frac{\sigma}{\gamma}\frac{1}{\sqrt{n}} W(\lambda(nt))\right|>r_n\right\}\leq \alpha_n,$ where $$r_n\to0$$, $$\alpha_n\to0$$ as $$n\to+\infty$$.

### MSC:

 60F17 Functional limit theorems; invariance principles 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60E15 Inequalities; stochastic orderings
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