Almost sure invariance principles for mixing sequences of random variables (Q1312302)

Let \((X_ n, n\geq 1)\) be a stationary mixing sequence of random variables with mean zero and finite variance; \(\rho(n)\) and \(\varphi(n)\) denote coefficients of \(\rho\)-mixing and uniform mixing, respectively. As usual, \(S_ n=X_ 1+\cdots+X_ n\), \(\sigma^ 2_ n=ES^ 2_ n\), \(n\geq 1\). It is established that if \(\sigma^ 2_ n@>>n\to\infty>\infty\) and the mixing rate satisfies \(\sum_ n\varphi^{1/2}(2^ n)<\infty\) or \(\rho(n)=O(\text{log}^{-r})\) for some \(r>1\), then \((S_ n, n\geq 1)\) could be redefined on a richer probability space on which there exists a standard Wiener process \((W(t), t\geq 0)\) such that \(S_ n-W(\sigma^ 2_ n)=o(\sigma_ n\sqrt{\text{log log} n})\) almost surely as \(n\to\infty\).