On the weak invariance principle for stationary sequences under projective criteria (Q867084)

Let \((\Omega,{\mathcal A}, P)\) be a probability space, and \(T: \Omega\to\Omega\) be a bijective bimeasurable transformation preserving the probability \(P\). Let \({\mathcal I}\) be the \(\sigma\)-algebra of all invariant sets. Let \({\mathcal M}_0\) be a \(\sigma\)-algebra of \({\mathcal A}\) satisfying \({\mathcal M}_0\subset T^{-1}({\mathcal M}_0)\). Put \({\mathcal M}_i= T^{-i}({\mathcal M}_0)\) and \({\mathcal M}_{-\infty}= \bigcap_{n\geq 0}{\mathcal M}_{-n}\). Let \(X_0\) be a \({\mathcal M}_0\)-measurable, centered real random variable such that \(EX^2_0<\infty\). Define \(X_i= X_0\circ T^i\) and set \(S_n= \sum^n_{k=1} X_k\) and \(W_n(t)= S_{[nt]}\). We assume that \(\sigma^2_n= \text{Var}(S_n)= nh(n)\) where \(h(n)\) is slowly varying in the strong sense and that there exists a positive \({\mathcal M}_{-\infty}\) random variable \(\eta\) such that \[ \lim_{n\to\infty}\, E\Biggl({S^2_n\over \sigma^2_n}\Biggl|{\mathcal M}_{-n}\Biggr)= \eta\qquad\text{in }L^1. \] In this paper, the authors prove among other things the following: (I) If \(S^2_n/\sigma^2_n\) is uniformly integrable and \(\| E(S_n|{\mathcal M}_{-n})\|_1= o(\sigma_n)\) \((n\to\infty)\), then \(\eta= \eta\circ T\) a.s. and \(\sigma^{-1}_n S_n\) converges in distribution to \(\sqrt{\eta}N\) where \(N\) is a standard Gaussian random variable independent of \({\mathcal I}\). (II) Let \(Q\) be the upper tail quantile function of the distribution \(Q_{X_0}\) of \(X_0\) and \(G\) be the inverse of the function \(H_y: x\to\int^x_0 Q_{X_0}(u)\,du\). If \[ n^2 \int^{n^{-1}\| E(S_n|{\mathcal M}_{-n})\|_1}_0 Q\circ G(u)\,du= o(\sigma^2_n)\quad (n\to\infty), \] then \((\sqrt{\pi/2}E|S_n|)^{-1} W_n\) converges in distribution to \(\sqrt{\eta}W\) where \(W\) is a standard Brownian motion independent of \({\mathcal I}\). Many interesting and important corollaries and examples are shown.