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By using the white noise theory for fractional Brownian sheet, we prove the stochastic Green’s theorem with respect to fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ ( 0 , 1 ) . As an application, we derive an Ito formula for fractional Brownian sheet with H 1 , H 2 ∈ ( 0 , 1 ) .

By applying the techniques of Malliavin calculus, we study a central limit theorem of cross-variation related to the standard Brownian sheet. Moreover, we provide the exact Berry–Esseen bound on the Kolmogorov distance by using a recent result proven by Nourdin and Peccati [Nourdin, I., & Peccati, G. (2009b). Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields. Annals of Probability, 37(6), 2231–2261].

비대체와 회귀대체는 조사변수의 모형과 조사변수와 보조변수의 관계에 의존하며 모형이 성립되지 않는 경우 이들 대체법을 이용한 추정량의 불편성은 보장되지 않는다. 본 연구에서는 모형이 성립되지 않는 경우에도 추정량의 근사적 불편성이 성립되는 로버스트 대체법을 개발한다. 대체법 개발시 보조변수의 모수 정보를 이용하여 추정량의 효율 증대를 가져오게 한다. 모의실험을 실시하여 본 연구에 대한 이론적 결과의 타당성을 보인다.

We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can be expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.

We present an extension of the Wong-Zakai type approximation theorem for a multiple stochastic integral. Using a piecewise linear approximation w (n) of a Wiener process w, we prove that the multiple integral process where f is a given symmetric function in the space 𝒞([0, T] m ), converge to the multiple Stratonovich integral of f in the uniform L 2-sense.

We present an extension of the Wong–Zakai approximation theorem for a stochastic differential equation on the plane driven by a two-parameter Wiener process. For an approximation of the two-parameter Wiener process, we use a two-parameter version of the one-parameter piecewise linear approximation. By our approximation to the two-parameter Wiener process we show that the solution of an ordinary differential equation converges, in the uniform L 2-sense, to that of a stochastic differential equation obtained by using Stratonovich integral. *This research was supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.