We consider a simple model for multidimensional conewise linear dynamics around cusplike equilibria. We assume that the local linear evolution is either v^{'}=Av or Bv (with A, B independently drawn from a rotationally invariant ensemble of symmetric N×N matrices) depending on the sign of the first component of v. We establish strong connections with the random diffusion persistence problem. When N→∞, we find that the Lyapunov exponent is non-self-averaging, i.e., one can observe apparent stability and apparent instability for the same system, depending on time and initial conditions. Finite N effects are also discussed and lead to cone trapping phenomena.