Laplace functional

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In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properties of metric spaces. One type of Laplace functional,[1][2] also known as a characteristic functional[a] is defined in relation to a point process, which can be interpreted as random counting measures, and has applications in characterizing and deriving results on point processes.[5] Its definition is analogous to a characteristic function for a random variable.

The other Laplace functional is for probability spaces equipped with metrics and is used to study the concentration of measure properties of the space.

Definition for point processes[edit]

For a general point process defined on , the Laplace functional is defined as:[6]

where is any measurable non-negative function on and

where the notation interprets the point process as a random counting measure; see Point process notation.


The Laplace functional characterizes a point process, and if it is known for a point process, it can be used to prove various results.[2][6]

Definition for probability measures[edit]

For some metric probability space (Xdμ), where (Xd) is a metric space and μ is a probability measure on the Borel sets of (Xd), the Laplace functional:

The Laplace functional maps from the positive real line to the positive (extended) real line, or in mathematical notation:


The Laplace functional of (Xdμ) can be used to bound the concentration function of (Xdμ), which is defined for r > 0 by


The Laplace functional of (Xdμ) then gives leads to the upper bound:


  1. ^ Kingman[3] calls it a "characteristic functional" but Daley and Vere-Jones[2] and others call it a "Laplace functional",[1][4] reserving the term "characteristic functional" for when is imaginary.


  1. ^ a b D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic geometry and its applications, volume 2. Wiley, 1995.
  2. ^ a b c D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, Springer, New York, second edition, 2003.
  3. ^ Kingman, John (1993). Poisson Processes. Oxford Science Publications. p. 28. ISBN 0-19-853693-3.
  4. ^ Baccelli, F. O. (2009). "Stochastic Geometry and Wireless Networks: Volume I Theory" (PDF). Foundations and Trends in Networking. 3 (3–4): 249–449. doi:10.1561/1300000006.
  5. ^ Barrett J. F. The use of characteristic functionals and cumulant generating functionals to discuss the effect of noise in linear systems, J. Sound & Vibration 1964 vol.1, no.3, pp. 229-238
  6. ^ a b F. Baccelli and B. B{\l}aszczyszyn. Stochastic Geometry and Wireless Networks, Volume I - Theory, volume 3, No 3-4 of Foundations and Trends in Networking. NoW Publishers, 2009.
  • Ledoux, Michel (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs. Vol. 89. Providence, RI: American Mathematical Society. pp. x+181. ISBN 0-8218-2864-9. MR1849347