  Statistical early warning signs can be used to identify an approaching
bifurcation in stochastic dynamical systems and are now regularly employed in
applications concerned with the identification of potential rapid, non-linear
change or tipping points. However, the reliability of these early warning signs
relies on a number of key mathematical assumptions, most notably the presence
of Gaussian noise. We here show that for systems driven by non-Gaussian,
$\alpha$-stable noise, the classical early warning signs of rising variance and
autocorrelation are not supported by mathematical theory and their use poses
the danger of spurious, false-positive results. To address this, we provide a
generalized approach by introduce the scaling factor $\gamma_X$ as an
alternative early warning sign. We show that in the case of the
Ornstein-Uhlenbeck process, there exists a direct inverse relationship between
$\gamma_{X}$ and the bifurcation parameter, telling us that $\gamma_{X}$ will
increase as we approach the bifurcation. Our numerical simulations confirm
theoretical results and show that our findings generalize well to non-linear,
non-equilibrium systems. We thus provide a generalized, robust and applicable
statistical early warning sign for systems driven by Gaussian and non-Gaussian
$\alpha$-stable noise.
