On the invariance principle and the law of iterated logarithm for stationary processes (Q2712785)

Let \(T\) be a one-to-one bimeasurable and measure preserving transformation of a probability space \((\Omega,{\mathcal A}, \mu)\) where the measure \(\mu\) is ergodic with respect to \(T\). Consider measurable functions \(f\) and \(g\) on \(\Omega\) such that \( f = m + g - g \circ T\). For the process \((m \circ T^i)\) assume the validity of the weak invariance principle (IP), respectively, the LIL (or the FLIL). Then the same holds true for the process \((f \circ T^i)\) iff \(n^{-1/2} \max_{1 \leq k \leq n} |g \circ T^k|\to 0\) in probability \((n \to \infty)\), respectively iff \((n \log \log n)^{-1/2} g \circ T^n \to 0\) a.s. Suppose \(0 < p < 2 \leq r\), \(g \in L^p\), \(g - g \circ T \in L^r\). If \(p \geq (r+2)/r\), then the IP and the LIL take place. Counterexamples are given for \(p < (r-1)/(r -3/2)\). A brief survey of related conditions is provided.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00063].