4 We construct a nearest-neighbor process ${S_n}$ on 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 $Ck^{-\infty}$ 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 $d \geq 4$, 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 Grimmett, Kesten
and Zhang, who proved that supercritical percolation clusters in $\mathbf{Z}^d$
are transient for all $d \geq 3$.