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Other kinds of distributions

Absolutely continuous and discrete distributions with support on R k {\displaystyle \mathbb {R} ^{k}} or N k {\displaystyle \mathbb {N} ^{k}} are extremely useful to model a myriad of phenomena, since most practical distributions are supported on relatively simple subsets, such as hypercubes or balls. However, this is not always the case, and there exist phenomena with supports that are actually complicated curves γ: → R n {\displaystyle \gamma :\rightarrow \mathbb {R} ^{n}} within some space R n {\displaystyle \mathbb {R} ^{n}} or similar. In these cases, the probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it.

One example is shown in the figure to the right, which displays the evolution of a system of differential equations (commonly known as the Rabinovich–Fabrikant equations) that can be used to model the behaviour of Langmuir waves in plasma. When this phenomenon is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system.

This kind of complicated support appears quite frequently in dynamical systems. It is not simple to establish that the system has a probability measure, and the main problem is the following. Let t 1 ≪ t 2 ≪ t 3 {\displaystyle t_{1}\ll t_{2}\ll t_{3}} be instants in time and O {\displaystyle O} a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set O {\displaystyle O} would be equal in interval {\displaystyle } and {\displaystyle }, which might not happen; for example, it could oscillate similar to a sine, sin (t) {\displaystyle \sin(t)}, whose limit when t → ∞ {\displaystyle t\rightarrow \infty } does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies the existence of a probability measure is ergodic theory.

Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively.