Point process notation: Difference between revisions

Revision as of 12:08, 12 October 2013

Revision as of 12:14, 12 October 2013

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==Interpretation of point processes==

The notation, as well as the terminology, of point processes depends on their setting and interpretation as mathematical objects which under certain assumptions can be interpreted as a random [[sequences]] of points, random [[Set (mathematics)|sets]] of points or random [[counting measure]]s<ref name="stoyan1995stochastic"/>.

===Random sequences of points===

In some mathematical frameworks, a given point process may be considered as a sequence of points with each point randomly positioned in ''d''-dimensional [[Euclidean space]] '''R'''<mathsup> \textbf{R}^d</mathsup><ref name="stoyan1995stochastic"/> as well as some other more abstract [[mathematical space]]s. In general, whether or not a random sequence is equivalent to the other interpretations of a point process depends on the underling mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''<mathsup> \textbf{R}^d</mathsup><ref name="daleyPPII2008"> D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. {II''}. Probability and its Applications (New York). Springer, New York, second edition, 2008.

</ref>.

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</ref> and the techniques of random measure theory offering another way to study point processes<ref name="stoyan1995stochastic"/><ref name="grandell1977point"> J. Grandell. Point processes and random measures. ''Advances in Applied Probability'', pages 502--526, 1977.

</ref>, which also induces the use of the various notations used in [[Integral#Terminology_and_notation|integration]] and measure theory. {{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mechke<ref name="stoyan1995stochastic"/>, varying [[integral]] notation in general applies to all integrals here and elsewhere.}}

==Dual notation==

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==Sums==

If <math> f</math> is some [[measurable function]] on '''R'''<mathsup> \textbf{R}^d</mathsup>, then the sum of <math> f(x)</math> over all the points <math> x</math> in <math> \Phi</math> can<ref name="stoyan1995stochastic"/> be written as:

:<math> f(x_1) + f(x_2) \dots </math>