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

This article is cited in 17 scientific papers (total in 17 papers)

The invariance principle for stationary processes

Yu. A. Davydov

Leningrad

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:

Received: 22.02.1968

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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  • Теория вероятностей и ее применения Theory of Probability and its Applications
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