We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space *C* [0, *T*]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation.

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Correspondence to B. V. Bondarev.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 6, pp. 733–753, June, 2010.

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Bondarev, B.V., Kozyr’, S.M. Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I.
*Ukr Math J* **62, **847–871 (2010). https://doi.org/10.1007/s11253-010-0395-6

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### Keywords

- Random Process
- Stochastic Differential Equation
- Independent Random Variable
- Wiener Process
- Invariance Principle