In probability theory, an isotropic measure is any mathematical measure that is invariant under linear isometries. It is a standard simplification and assumption used in probability theory. Generally, it is used in the context of measure theory on -dimensional Euclidean space, for which it can be intuitive to study measures that are unchanged by rotations and translations. An obvious example of such a measure is the standard way of assigning a measure to subsets of n-dimensional Euclidean space: Lebesgue measure.

Definition

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An isotropic measure on   is a (Borel) measure that is absolutely continuous on   and that is invariant under linear isometries of  .[1] Alternatively, an isotropic measure,  , is a measure for which there exists a real density function   on   such that   for  .[2]

Example

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  • The Lebesgue measure on   is invariant under linear isometries and is hence an isotropic measure. In this case,  .
  • For  , the linear isometries of   are of the form   or  , for some constant  . Hence an isotropic measure on   must satisfy  , for any   and  . The measure  , for  , is one such isotropic measure.

Unimodal measure

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In probability theory it is common that another assumption is added to measures in addition to the measure being isotropic. A unimodal measure (or isotropic unimodal measure) is any isotropic measure   such that   is nonincreasing on  . It is possible that  .[2]

Isotropic and unimodal stochastic processes

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In studying stochastic processes, in particular Lévy processes,[3] a reasonable assumption to make is that, for each element of the index set, the probability distributions of the random variables are isotropic or even unimodal measures.

More specifically, an isotropic Lévy process is a Lévy process,  , such that all its distributions,  , are isotropic measures.[1] A unimodal Lévy process (or isotropic unimodal Lévy process) is a Lévy process,  , such that all its distributions,  , are unimodal measures.[1]

See also

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References

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  1. ^ a b c Bogdan, Krzysztof; Grzywny, Tomasz; Ryznar, Michał (2014-06-07). "Barriers, exit time and survival probability for unimodal Lévy processes". Probability Theory and Related Fields. 162 (1–2): 155–198. arXiv:1307.0270. doi:10.1007/s00440-014-0568-6. ISSN 0178-8051.
  2. ^ a b Toshiro, Watanabe (1983). "The isoperimetric inequality for isotropic unimodal Lévy processes". Z. Wahrsch. Verw. Gebiete. 63 (4): 487–499.
  3. ^ Sato, Ken-iti (1999-01-01). Lévy processes and infinitely divisible distributions. Cambridge University Press. ISBN 978-0521553025. OCLC 41142930.