  Signal-to-interference plus noise ratio (SINR) percolation is an
infinite-range dependent variant of continuum percolation modeling connections
in a telecommunication network. Unlike in earlier works, in the present paper
the transmitted signal powers of the devices of the network are assumed random,
i.i.d. and possibly unbounded. Additionally, we assume that the devices form a
stationary Cox point process, i.e., a Poisson point process with stationary
random intensity measure, in two or higher dimensions. We present the following
main results. First, under suitable moment conditions on the signal powers and
the intensity measure, there is percolation in the SINR graph given that the
device density is high and interferences are sufficiently reduced, but not
vanishing. Second, if the interference cancellation factor $\gamma$ and the
SINR threshold $\tau$ satisfy $\gamma \geq 1/(2\tau)$, then there is no
percolation for any intensity parameter. Third, in the case of a Poisson point
process with constant powers, for any intensity parameter that is supercritical
for the underlying Gilbert graph, the SINR graph also percolates with some
small but positive interference cancellation factor.
