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In [[mathematics]] — specifically, in [[large deviations theory]] — the '''contraction principle''' is a [[theorem]] that states how a large deviation principle on one space "[[push forward|pushes forward]]" to a large deviation principle on another space ''via'' a [[continuous function]]. | |||
==Statement of the theorem== | |||
Let ''X'' and ''Y'' be [[Hausdorff space|Hausdorff]] [[topological space]]s and let (''μ''<sub>''ε''</sub>)<sub>''ε''>0</sub> be a family of [[probability measure]]s on ''X'' that satisfies the large deviation principle with [[rate function]] ''I'' : ''X'' → [0, +∞]. Let ''T'' : ''X'' → ''Y'' be a continuous function, and let ''ν''<sub>''ε''</sub> = ''T''<sub>∗</sub>(''μ''<sub>''ε''</sub>) be the [[Pushforward measure|push-forward measure]] of ''μ''<sub>''ε''</sub> by ''T'', i.e., for each [[measurable set]]/event ''E'' ⊆ ''Y'', ''ν''<sub>''ε''</sub>(''E'') = ''μ''<sub>''ε''</sub>(''T''<sup>−1</sup>(''E'')). Let | |||
:<math>J(y) := \inf \big\{ I(x) \big| x \in X \mbox{ and } T(x) = y \big\},</math> | |||
with the convention that the [[infimum]] of ''I'' over the [[empty set]] ∅ is +∞. Then: | |||
* ''J'' : ''Y'' → [0, +∞] is a rate function on ''Y'', | |||
* ''J'' is a good rate function on ''Y'' if ''I'' is a good rate function on ''X'', and | |||
* (''ν''<sub>''ε''</sub>)<sub>''ε''>0</sub> satisfies the large deviation principle on ''Y'' with rate function ''J''. | |||
==References== | |||
* {{cite book | |||
| last= Dembo | |||
| first = Amir | |||
| coauthors = Zeitouni, Ofer | |||
| title = Large deviations techniques and applications | |||
| series = Applications of Mathematics (New York) 38 | |||
| edition = Second edition | |||
| publisher = Springer-Verlag | |||
| location = New York | |||
| year = 1998 | |||
| pages = xvi+396 | |||
| isbn = 0-387-98406-2 | |||
| mr = 1619036 | |||
}} (See chapter 4.2.1) | |||
* {{cite book | |||
| last = den Hollander | |||
| first = Frank | |||
| title = Large deviations | |||
| series = [[Fields Institute]] Monographs 14 | |||
| publisher = [[American Mathematical Society]] | |||
| location = Providence, RI | |||
| year = 2000 | |||
| pages = x+143 | |||
| isbn = 0-8218-1989-5 | |||
| mr = 1739680 | |||
}} | |||
[[Category:Asymptotic analysis]] | |||
[[Category:Large deviations theory]] | |||
[[Category:Mathematical principles]] | |||
[[Category:Probability theorems]] | |||
Revision as of 06:26, 21 March 2013
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" to a large deviation principle on another space via a continuous function.
Statement of the theorem
Let X and Y be Hausdorff topological spaces and let (με)ε>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let T : X → Y be a continuous function, and let νε = T∗(με) be the push-forward measure of με by T, i.e., for each measurable set/event E ⊆ Y, νε(E) = με(T−1(E)). Let
with the convention that the infimum of I over the empty set ∅ is +∞. Then:
- J : Y → [0, +∞] is a rate function on Y,
- J is a good rate function on Y if I is a good rate function on X, and
- (νε)ε>0 satisfies the large deviation principle on Y with rate function J.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (See chapter 4.2.1) - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534