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The invariance principle for weakly dependent variables

В. A. Lifšic

Leningrad

Abstract: Let $S_n=X_{n1}+\dots+X_{nn}$, $\mathbf DX_{nk}<\infty$, $\mathbf EX_{nk}=0$. Denote $\mathscr F_k=\mathscr F_{nk}=\sigma\{(X_{ns})_{s\ge k}\}$ and $E_kZ=\mathbf E(Z\mid\mathscr F_k)$. Let $\sigma$-field $\mathscr E^k=\sigma\{(E_j1_{X_{ni}<q})_{i\le j\le k,\,q\in R}\}$,
\begin{gather*} \gamma_n(r)=\sup_k\sup_{B\in\mathscr F_{k+r}}\sup_{A_1,A_2\in\mathscr E^k} |\mathbf P(B\mid A_1)- \mathbf(B\mid A_2)|, \\ l_n=\min_{m\ge 2}\biggl(1+\sum_{n/m>r\ge 1}\sqrt{\gamma_n(mr)}\biggr)^{1/2}\biggl(m+\sum_{r\ge 1}\gamma_n(r)\biggr),\quad B_n^2=\mathbf DS_n. \end{gather*}
We define the random functions on $[0, 1]$
$$ \xi_n(t)=B_n^{-1}\sum_{j\ge 1}X_{nj}\mathbf 1_{b_j\le tB_n^2},\qquad b_j=(\mathbf D-\mathbf DE_j)\sum_{k=1}^jX_{nk}, $$
and denote by $\mathscr L(\xi_n)$ the distribution of $\xi_n$ in the Skorohod space.
Theorem. {\it If $\displaystyle\lim_{n\to\infty}B_n^{-2}\biggl(l_n+\sum_{r=1}^{n-1}\sqrt{\gamma_n(r)}\biggr)\sum_{j=1}^n\mathbf EX_{nj}^2 1_{|X_{nj}|>\varepsilon B_n/l_n}=0$ for every $\varepsilon>0$, then $\mathscr L(\xi_n)$ converges weakly to a Wiener distribution.}
The estimate $\displaystyle\mathbf DS_n\ge\frac{1}{16}(1-\gamma_n(1))\sum_{k=1}^n\mathbf DX_{nk}$ is obtained also.
This theorem generalizes the well-known Dobrusin's results [9] for inhomogeneous Markow chains.

Received: 12.03.1980

 English version: Theory of Probability and its Applications, 1985, 29:1, 33–40

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