Summary: We construct a nearest-neighbor process

$\left\{{S}_{n}\right\}$ on

$\mathbf{Z}$ that is less predictable than simple random walk, in the sense that given the process until time

$n$, the conditional probability that

${S}_{n+k}=x$ is uniformly bounded by

$C{k}^{-\alpha}$ for some

$\alpha >1/2$. From this process, we obtain a probability measure

$\mu $ on oriented paths in

${\mathbf{Z}}^{3}$ such that the number of intersections of two paths, chosen independently according to

$\mu $, has an exponential tail. (For

$d\ge 4$, the uniform measure on oriented paths from the origin in

${\mathbf{Z}}^{d}$ has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter

$p$ is close enough to 1. This yields an extension of a theorem of

*G. R. Grimmet*,

*H. Kesten* and

*Y. Zhang* [Probab. Theory Relat. Fields 96, No. 1, 33-44 (1993;

Zbl 0791.60095)], who proved that supercritical percolation clusters in

${\mathbf{Z}}^{d}$ are transient for all

$d\ge 3$.