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

Bondarev, B. V.; Baev, A. V.

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 \(n\to+\infty\).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

Ornstein-Uhlenbeck process; distribution of supremum; Poisson process