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 Teor. Veroyatnost. i Primenen., 1970, Volume 15, Issue 3, Pages 498–509 (Mi tvp1857)

The invariance principle for stationary processes

Yu. A. Davydov

Abstract: Let $X_t$ be a stationary process, $\mathbf EX_t=0$, $\mathbf DX_t=\sigma^2<\infty$, $S_T=\sum_1^TX_t$ (discrete time), or $S_t=\int_0^TX_t dt$ (continuous time). Define $X_T(t)$ as in (1) or (2). Let $B_T(s,t)$ be the covariance function of $X_t(t)$. Let distribution $P_T$ correspond to the process $X_T(t)$ and distribution $W_\gamma$ correspond to a Gaussian process with the covariance function
$$B_\gamma(s,t)=\frac12(s+t+|s^{1/\gamma}-t^{1/\gamma}|^\gamma).$$

Theorem 1. If $\psi(T)=\mathbf DS_T\uparrow\infty$, $P_T\Rightarrow W_\gamma$, then $B_T(s,t)\to B_\gamma(s,t)$ and $\psi(T)=T^\gamma h(T)$, where $h(T)$ is a slowly changing function.
Theorem 2. {\em Let $X_j=\sum_{i=-\infty}^\infty c_{i-j}\xi_i$ where $\xi_i$ are independent identically distributed random variables, $\mathbf E\xi_i=0$, $\mathbf E\xi_i^{2k}<\infty$ and $\sum c_j^2<\infty$. If $\mathbf DS_n=n^\gamma h(n)$, $2/(k+2)<\gamma\le2$, where $h(n)$ is a slowly changing function, then $P_n\Rightarrow W_\gamma$.}
In the next two theorems the invariance principle is proved for processes generated by mixing processes.

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English version:
Theory of Probability and its Applications, 1970, 15:3, 487–498

Bibliographic databases:

Citation: Yu. A. Davydov, “The invariance principle for stationary processes”, Teor. Veroyatnost. i Primenen., 15:3 (1970), 498–509; Theory Probab. Appl., 15:3 (1970), 487–498

Citation in format AMSBIB
\Bibitem{Dav70} \by Yu.~A.~Davydov \paper The invariance principle for stationary processes \jour Teor. Veroyatnost. i Primenen. \yr 1970 \vol 15 \issue 3 \pages 498--509 \mathnet{http://mi.mathnet.ru/tvp1857} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=283872} \zmath{https://zbmath.org/?q=an:0209.48904} \transl \jour Theory Probab. Appl. \yr 1970 \vol 15 \issue 3 \pages 487--498 \crossref{https://doi.org/10.1137/1115050} 

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