I have two questions about the uniform integrability.The definition I am using is that a class of random variables χ is uniformly integrable if given an ϵ &gt; 0, there exists a k such that for any x in χ we have E [ | x | I { x ≥ k } ] &lt; ϵ.First, is that in the definition of uniform integrability, can the density function which we compute the expectation with respect to it, vary with n?Second, suppose I have a sequence of random variables ( x n ) n ≥ 1 (with varying density function w.r.t n). For the two functions f , g, I know that f ( x ) ≤ g ( x ) for all x. Does the uniform integrability of the sequence ( g ( x n ) ) n ≥ 1 imply the uniform integrability of ( f ( x n ) ) n ≥ 1 ?

Question

I have two questions about the uniform integrability.The definition I am using is that a class of random variables   χ is uniformly integrable if given an   ϵ  &amp;gt;  0, there exists a   k such that for any   x in   χ we have       E    [      |    x      |        I    {  x  ≥  k  }  ]  &amp;lt;  ϵ.First, is that in the definition of uniform integrability, can the density function which we compute the expectation with respect to it, vary with   n?Second, suppose I have a sequence of random variables   (      x    n        )          n      ≥      1       (with varying density function w.r.t n). For the two functions   f  ,  g, I know that   f  (  x  )  ≤  g  (  x  ) for all   x. Does the uniform integrability of the sequence   (  g  (      x    n    )      )          n      ≥      1       imply the uniform integrability of   (  f  (      x    n    )      )          n      ≥      1      ?

Ordettyreomqu · Accepted Answer

A family of random variables   (      X          α            )          α      ∈      A       is u.i. if      sup          α      ∈      A            E        |        X          α            |    1  {      |        X          α            |    &amp;gt;  M  }  →  0as   M  →  ∞. Note that       X          α      &#039;s may have different distributions (densities if exist). If   (      Y          α            )          α      ∈      A       is a u.i. family of r.v.s. satisfying       |        X          α            |    ≤      |        Y          α            |   a.s. for all   α  ∈  A, then   (      X          α            )          α      ∈      A       is u.i. as well because for any   α  ∈  A,      E        |        X          α            |    1  {      |        X          α            |    &amp;gt;  M  }  ≤      E        |        Y          α            |    1  {      |        Y          α            |    &amp;gt;  M  }  .