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Equivalence of renewal sequences and isomorphism of random walks. (English) Zbl 0808.60063
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 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 \(E\biggl(| X|^ 2 \sqrt{ \log^ + | X|}\biggr) < \infty\).
Reviewer: M.Quine (Sydney)

60G50 Sums of independent random variables; random walks
60K05 Renewal theory
Full Text: DOI
[1] J. Aaronson and M. Keane,Isomorphism of random walks, Israel J. Math., this issue. · Zbl 0804.60057
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