The time evolution of the climate system is unpredictable. The inherent unpredictability originates from the irregular, chaotic nature of the solutions of the underlying dynamics, and in such circumstances only probabilistic predictions can be made. The full range of possible outcomes is described by the so-called natural distribution on a dynamical attractor, since any initial probability distribution converges to the natural one with an exponential speed. We emphasize that this distribution exists, beyond what a traditional framework covers, in non-periodically time-dependent dynamical systems as well, so that this approach is applicable even if a parameter (like the greenhouse gas concentration) is shifting. In this case the natural distribution itself depends on time, and this dependence (especially shifts in expectation values) represents climate change. Numerically, the natural distribution can be represented by an ensemble of trajectories, and statistical characteristic are to be evaluated with respect to the ensemble. We demonstrate on the example of the so-called teleconnections that this proper approach can give considerably different results compared to the traditional one which evaluates statistics along a single trajectory over a finite interval of time. In general, differences between ensemble and temporal averages cannot be arbitrarily reduced in non-periodically time-dependent dynamical systems.
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