# zbMATH — the first resource for mathematics

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)

##### MSC:
 60G50 Sums of independent random variables; random walks 60K05 Renewal theory
Full Text:
##### References:
 [1] J. Aaronson and M. Keane,Isomorphism of random walks, Israel J. Math., this issue. · Zbl 0804.60057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.