Weak invariance principles for sums of dependent random functions
Let \(\{\varepsilon_j(t); 0\leq t\leq 1\}^\infty_{j=-\infty}\) be i.i.d. random functions with values in a mesurable space \({\mathcal S}\) such that \(\varepsilon_j(t)= \varepsilon_j(t,\omega)\) is jointly measurable in \((t,\omega)\). Let \(g:{\mathcal S}^\infty\to{\mathcal L}^2\) be a nonrandom measurable function. Let \(\mathbb{X}= \{X_i(t)\}^\infty_{-\infty}\) be a sequence of Bernoulli shifts, that is, \(X_j(t)= g(\varepsilon_j(t), \varepsilon_{j-1}(t),\dots)\) \((-\infty< j<\infty)\). Let \(\|\cdot\|\) denote the \(L^2[0,1]\) norm. Suppose \(\operatorname{E}X_0= 0\) for all \(0\leq t\leq 1\) and \(\operatorname{E}\| X_0\|^{2+\delta}<\infty\) for some \(0<\delta< 1\). Suppose, further, that the sequence \(\{X_n\}^\infty_{n=-\infty}\) can be approximated by \(l\)-dependent sequences \(\{X_{n,l}\}^\infty_{n=-\infty}\) in the sense \[ \sum^\infty_{l=1} (\operatorname{E}\| X_n- X_{n,l}\|^{2+ \delta})^{{1\over\kappa}}< \infty\quad\text{for some }\kappa> 2+ \delta, \] where \(X_{n,l}= g(\varepsilon_n, \varepsilon_{n-1},\dotsc, \varepsilon_{n-l+1}, \varepsilon^*_{n,l})\), \(\varepsilon^*_{n,l}= (\varepsilon^*_{n,l,n-l}, \varepsilon^*_{n,l,n-l-1},\dots)\), the \(\varepsilon^*_{n,l,k}\) are independent copies of \(\varepsilon_0\), independent of \(\{\varepsilon_i: -\infty< i<\infty\}\). Since the series in \[ C(t,s)= \operatorname{E}X_0(t) X_0(s)+ \sum^\infty_{l=1} \operatorname{E}X_0(t) X_l(s)+ \sum^\infty_{l=1} \operatorname{E}X_0(s) X_l(t) \] are convergent and \(C(s, t)\) is positive-definite, there exists \(\lambda_1\geq\lambda_2\geq\cdots\geq 0\) and orthonormal functions \(\phi_i(t)\) \((0\leq t\leq 1)\) satisfying \[ \lambda_i \phi_i(t)= \in^1_0 C(t,s) \phi_i(s)\,ds\qquad (1\leq i<\infty). \] Define a Gaussian process \[ \Gamma(x,t)= \sum^\infty_{i=1} \lambda^{{1\over 2}}_i W_i(x) \phi_i(t), \] where the \(W_i\)'s are i.i.d. Wiener processes. In this paper, the authors prove that, for every \(N\), we can define a Gaussian process \(\Gamma_N(x, t)\) \((=^{{\mathcal D}}\{\Gamma(x,t): 0\leq x,t\leq 1\})\) such that \[ \sup_{0\leq x\leq 1}\,\int^1_0 \Biggl({1\over \sqrt{N}}\, \sum^{[Nx]}_{i=1} X_i(t)- \Gamma_N(x,t)\Biggr)^2\,dt= o_p(1) \] holds.
scientific article; zbMATH DE number 6128617
scientific article