Equivalence of renewal sequences and isomorphism of random walks (Q1335146)

The random walk with jump random variable \(X \in \mathbb{Z}^ d\) is a measure preserving transformation \(T_ X\) defined by \[ T_ X((\dots, x_{-1}, x_ 0, x_ 1, \dots), n) = ((\dots, x_ 0, x_ 1, x_ 2,\dots), n + x_ 1). \] The random variable \(Y\) takes values \(\{0, \pm 1\}^ d\) with probability \(3^{-d}\). It is shown by the first author and \textit{M. Keane} [ibid. 87, No. 1/3, 37-63 (1994; Zbl 0804.60057)] that for \(d = 1,2\), if \(X\) has zero mean and is strictly aperiodic (in the sense that its characteristic function \(\varphi\) satisfies \(| \varphi(s)| = 1\) if and only if \(s \in 2\pi \mathbb{Z}^ d\)), then a sufficient condition for \(T_ X\) and \(T_ Y\) to be isomorphic is \(E(| X|^ 7) < \infty\). Here this moment condition is reduced to \(E\biggl(| X|^ 2 \sqrt{ \log^ + | X|}\biggr) < \infty\).