  Warning signs for tipping points (or critical transitions) have been very
actively studied. Although the theory has been applied successfully in models
and in experiments for many complex systems such as for tipping in climate
systems, there are ongoing debates, when warning signs can be extracted from
data. In this work, we shed light on this debate by considering different types
of underlying noise. Thereby, we significantly advance the general theory of
warning signs for nonlinear stochastic dynamics. A key scenario deals with
stochastic systems approaching a bifurcation point dynamically upon slow
parameter variation. The stochastic fluctuations are generically able to probe
the dynamics near a deterministic attractor to reveal critical slowing down.
Using scaling laws near bifurcations, one can then anticipate the distance to a
bifurcation. Previous warning signs results assume that the noise is Markovian,
most often even white. Here, we study warning signs for non-Markovian systems
including coloured noise and $\alpha$-regular Volterra processes (of which
fractional Brownian motion and the Rosenblatt process are special cases). We
prove that early warning scaling laws can disappear completely or drastically
change their exponent based upon the parameters controlling the noise process.
This provides a clear explanation, why applying standard warning signs results
to reduced models of complex systems may not agree with data-driven studies. We
demonstrate our results numerically in the context of a box model of the
Atlantic Meridional Overturning Circulation (AMOC).
