Uniform integrability

  (Redirected from Uniformly integrable)

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.

Measure-theoretic definitionEdit

Textbooks on real analysis and measure theory often use the following definition.[1][2]

Let   be a positive measure space. A set   is called uniformly integrable if to each   there corresponds a   such that

 

whenever   and  

Probability definitionEdit

In the theory of probability, the following definition applies.[3][4][5]

  • A class   of random variables is called uniformly integrable (UI) if given  , there exists   such that  , where   is the indicator function  
  • An alternative definition involving two clauses may be presented as follows: A class   of random variables is called uniformly integrable if:
    • There exists a finite   such that, for every   in  ,   and
    • For every   there exists   such that, for every measurable   such that   and every   in  ,  .

The two probabilistic definitions are equivalent.[6]

Relationship between definitionsEdit

The two definitions are closely related. A probability space is a measure space with total measure 1. A random variable is a real-valued measurable function on this space, and the expectation of a random variable is defined as the integral of this function with respect to the probability measure.[7] Specifically,

Let   be a probability space. Let the random variable   be a real-valued  -measurable function. Then the expectation of   is defined by

 
provided that the integral exists.

Then the alternative probabilistic definition above can be rewritten in measure theoretic terms as: A set   of real-valued functions is called uniformly integrable if:

  • There exists a finite   such that, for every   in  ,  .
  • For every   there exists   such that, for every measurable   such that   and for every   in  ,  .

Comparison of this definition with the measure theoretic definition given above shows that the measure theoretic definition requires only that each function be in  . In other words,   is finite for each  , but there is not necessarily an upper bound to the values of these integrals. In contrast, the probabilistic definition requires that the integrals have an upper bound.

One consequence of this is that uniformly integrable random variables (under the probabilistic definition) are tight. That is, for each  , there exists   such that

 
for all  .[8]

In contrast, uniformly integrable functions (under the measure theoretic definition) are not necessarily tight.[9]

In his book, Bass uses the term uniformly absolutely continuous to refer to sets of random variables (or functions) which satisfy the second clause of the alternative definition. However, this definition does not require each of the functions to have a finite integral.[10] The term "uniform absolute continuity" is not standard, but is used by some other authors.[11][12]

Related corollariesEdit

The following results apply to the probabilistic definition.[13]

  • Definition 1 could be rewritten by taking the limits as
     
  • A non-UI sequence. Let  , and define
     
    Clearly  , and indeed   for all n. However,
     
    and comparing with definition 1, it is seen that the sequence is not uniformly integrable.
 
Non-UI sequence of RVs. The area under the strip is always equal to 1, but   pointwise.
  • By using Definition 2 in the above example, it can be seen that the first clause is satisfied as   norm of all  s are 1 i.e., bounded. But the second clause does not hold as given any   positive, there is an interval   with measure less than   and   for all  .
  • If   is a UI random variable, by splitting
     
    and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in  .
  • If any sequence of random variables   is dominated by an integrable, non-negative  : that is, for all ω and n,
     
    then the class   of random variables   is uniformly integrable.
  • A class of random variables bounded in   ( ) is uniformly integrable.

Relevant theoremsEdit

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  .

  • DunfordPettis theorem[14][15]
    A class of random variables   is uniformly integrable if and only if it is relatively compact for the weak topology  .
  • de la Vallée-Poussin theorem[16][17]
    The family   is uniformly integrable if and only if there exists a non-negative increasing convex function   such that
     

Relation to convergence of random variablesEdit

A sequence   converges to   in the   norm if and only if it converges in measure to   and it is uniformly integrable. In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable.[18] This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.

CitationsEdit

  1. ^ Rudin, Walter (1987). Real and Complex Analysis (3 ed.). Singapore: McGraw–Hill Book Co. p. 133. ISBN 0-07-054234-1.
  2. ^ Royden, H.L. & Fitzpatrick, P.M. (2010). Real Analysis (4 ed.). Boston: Prentice Hall. p. 93. ISBN 978-0-13-143747-0.
  3. ^ Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5.
  4. ^ Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 214–218. ISBN 0-387-22833-0.
  5. ^ Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356–357. ISBN 978-1-107-00800-7.
  6. ^ Gut 2005, p. 214.
  7. ^ Bass 2011, p. 348.
  8. ^ Gut 2005, p. 236.
  9. ^ Royden and Fitzpatrick 2010, p. 98.
  10. ^ Bass 2011, p. 356.
  11. ^ Benedetto, J. J. (1976). Real Variable and Integration. Stuttgart: B. G. Teubner. p. 89. ISBN 3-519-02209-5.
  12. ^ Burrill, C. W. (1972). Measure, Integration, and Probability. McGraw-Hill. p. 180. ISBN 0-07-009223-0.
  13. ^ Gut 2005, pp. 215–216.
  14. ^ Dunford, Nelson (1938). "Uniformity in linear spaces". Transactions of the American Mathematical Society. 44 (2): 305–356. doi:10.1090/S0002-9947-1938-1501971-X. ISSN 0002-9947.
  15. ^ Dunford, Nelson (1939). "A mean ergodic theorem". Duke Mathematical Journal. 5 (3): 635–646. doi:10.1215/S0012-7094-39-00552-1. ISSN 0012-7094.
  16. ^ Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
  17. ^ Poussin, C. De La Vallee (1915). "Sur L'Integrale de Lebesgue". Transactions of the American Mathematical Society. 16 (4): 435–501. doi:10.2307/1988879. hdl:10338.dmlcz/127627. JSTOR 1988879.
  18. ^ Bogachev, Vladimir I. (2007). Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN 978-3-540-34513-8.

ReferencesEdit