Uniform integrability

Uniform integrability (Redirected from Dunford–Pettis theorem) Aller à la navigation Aller à la recherche En mathématiques, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

Contenu 1 Measure-theoretic definition 2 Probability definition 3 Tightness and uniform integrability 4 Related corollaries 5 Relevant theorems 6 Relation to convergence of random variables 7 Citations 8 References Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in {displaystyle L_{1}} which is central in dominated convergence. Several textbooks on real analysis and measure theory often use the following definition:[1][2] Definition A: Laisser {style d'affichage (X,{mathfrak {M}},dans )} be a positive measure space. A set {displaystyle Phi subset L^{1}(dans )} is called uniformly integrable if {displaystyle sup _{fin Phi }|F|_{L_{1}(dans )}0} there corresponds a {displaystyle delta >0} such that {displaystyle int _{E}|f|,dmu g}}|F|,dmu =0} où {displaystyle L_{+}^{1}(dans )={gin L^{1}(dans ):ggeq 0}} .

For finite measure spaces the following result[4] follows from Definition H: Théorème 1: Si {style d'affichage (X,{mathfrak {M}},dans )} est un (positif) finite measure space, then a set {displaystyle Phi subset L^{1}(dans )} is called uniformly integrable if and only if {displaystyle inf _{ageq 0}souper _{fin Phi }entier _{{|F|>a}}|F|,dmu =0} May textbooks in probability present Theorem 1 as the definition of uniform integrability in Probability spaces. When the space {style d'affichage (X,{mathfrak {M}},dans )} est {style d'affichage sigma } -fini, Definition H yields the following equivalency: Théorème 2: Laisser {style d'affichage (X,{mathfrak {M}},dans )} être un {style d'affichage sigma } -finite measure space, et {displaystyle hin L^{1}(dans )} be such that {displaystyle h>0} presque sûrement. A set {displaystyle Phi subset L^{1}(dans )} is called uniformly integrable if and only if {displaystyle sup _{fin Phi }|F|_{L_{1}(dans )}0} , there exits {displaystyle delta >0} tel que {displaystyle sup _{fin Phi }entier _{UN}|F|,dmu 0} il existe {displaystyle delta >0} tel que, for every measurable {style d'affichage A} tel que {style d'affichage P(UN)leq delta } et chaque {style d'affichage X} dans {style d'affichage {mathématique {C}}} , {nom de l'opérateur de style d'affichage {E} (|X|JE_{UN})leq varepsilon } .

or alternatively 2. A class {style d'affichage {mathématique {C}}} of random variables is called uniformly integrable (UI) if there exists {displaystyle Kin [0,infime )} tel que {nom de l'opérateur de style d'affichage {E} (|X|JE_{|X|geq K})leq varepsilon {texte{ for all X}}dans {mathématique {C}}} , où {style d'affichage I_{|X|geq K}} is the indicator function {style d'affichage I_{|X|geq K}={commencer{cas}1&{texte{si }}|X|geq K,\0&{texte{si }}|X|0} , il existe {displaystyle a>0} tel que {style d'affichage P(|X|>a)leq delta } pour tous {displaystyle Xin {mathématique {C}}} .[8] This however, does not mean that the family of measures {style d'affichage {mathématique {V}}_{mathématique {C}}:={Gros {}dans _{X}:Amapsto int _{UN}|X|,dP,,Xin {mathématique {C}}{Gros }}} is tight.

There is another notion of uniformity, slightly different than uniform integrability, which also has many applications in Probability and measure theory, and which does not require random variables to have a finite integral[9] Définition: Supposer {style d'affichage (Oméga ,{mathématique {F}},P)} is a probability space. A classed {style d'affichage {mathématique {C}}} of random variables is uniformly absolutely continuous with respect to {style d'affichage P} if for any {displaystyle varepsilon >0} , there is {displaystyle delta >0} tel que {style d'affichage E[|X|JE_{UN}]K)+nom de l'opérateur {E} (|X|,|X|1} ) is uniformly integrable. Relevant theorems In the following we use the probabilistic framework, but regardless of the finiteness of the measure, by adding the boundedness condition on the chosen subset of {displaystyle L^{1}(dans )} .

Dunford–Pettis theorem[13][14] A class of random variables {style d'affichage X_{n}subset L^{1}(dans )} is uniformly integrable if and only if it is relatively compact for the weak topology {style d'affichage sigma (L^{1},L^{infime })} . de la Vallée-Poussin theorem[15][16] The family {style d'affichage {X_{alpha }}_{alpha in mathrm {UN} }subset L^{1}(dans )} is uniformly integrable if and only if there exists a non-negative increasing convex function {style d'affichage G(t)} tel que {style d'affichage lim _{tto infty }{frac {g(t)}{t}}=infty {texte{ et }}souper _{alpha }nom de l'opérateur {E} (g(|X_{alpha }|))

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