In probability and statistics, a moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. Moment measures generalize the idea of (raw) moments of random variables, hence arise often in the study of point processes and related fields.[1]
An example of a moment measure is the first moment measure of a point process, often called mean measure or intensity measure, which gives the expected or average number of points of the point process being located in some region of space.[2] In other words, if the number of points of a point process located in some region of space is a random variable, then the first moment measure corresponds to the first moment of this random variable.[3]
Moment measures feature prominently in the study of point processes[1][4][5] as well as the related fields of stochastic geometry[3] and spatial statistics[5][6] whose applications are found in numerous scientific and engineering disciplines such as biology, geology, physics, and telecommunications.[3][4][7]
^ abD. 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.
^F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume I – Theory, volume 3, No 3-4 of Foundations and Trends in Networking. NoW Publishers, 2009.
^ abcD. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Stochastic geometry and its applications, volume 2. Wiley Chichester, 1995.
^ abD. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. I. Probability and its Applications (New York). Springer, New York, second edition, 2003.
^ abA. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications. Stochastic Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004, pages 1-75, 2007.
^J. Moller and R. P. Waagepetersen. Statistical inference and simulation for spatial point processes. CRC Press, 2003.
^F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume II – Applications, volume 4, No 1-2 of Foundations and Trends in Networking. NoW Publishers, 2009.
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