Invariance principles under weak dependence
Let \(X_ n\), \(n\geq 1\) be a sequence of random variables with \(EX_ i=0\) for every i. Denote by \(F^ m_ n=\sigma (X_ i,n\leq i<m)\), \(\| X\|_ p\) the norm in \(L_ p\), \(S_ k(n)=\sum^{k+n}_{i=k+1}X_ i\), \(\sigma_ n=\| S_ 0(n)\|_ 2\), \(W_ n(t)=S_ 0([nt])/\sigma_ n\), where \(t\in [0,1]\), \(X_ 0=0\), \[ \alpha_ p=\sup_{q\in N}\sup_{A\in F^ q_ 0,B\in F^{\infty}_{p+q}}| P(A\cap B)-P(A)P(B)|, \] \[ \rho_ p=\sup_{q\in N}\sup_{X\in L_ 2(F^ q_ 0),Y\in L_ 2(F^{\infty}_{p+q})}| Corr(X,Y)|, \rho^*=\lim_{p\to \infty}\rho_ p. \] For every \(0\leq \delta \leq 1\) denote \(f_{2+\delta}=2+2(1+\delta)\rho^{*2/(2+\delta)}+\delta (1+\delta)\rho^{*2\delta /(2+\delta)}\). If \(X_ n\), \(n\geq 1\) is a strong mixing sequence \((\alpha_ n\to 0\), \(n\to \infty)\) second order stationary satisfying: (i) \(\sigma_ n\to \infty\), \(n\to \infty\); (ii) for some \(\epsilon >0\) \[ \max_{1\leq i\leq n}\| X_ i\|_{2+\delta}=o(\sigma_ nn^{-(\log f_{2+\delta})/(2+\delta)- \epsilon}),\quad 0<\delta \leq 1, \] then 1) \(W_ n\) converges to the standard Brownian process W; 2) \(E(| S_ 0(n)| /\sigma_ n)^{2+\delta}\to \beta_{2+\delta}\), where \(\beta_{2+\delta}\) is the \(2+\delta\) absolute moment of N(0,1).
scientific article; zbMATH DE number 3973952