I have two questions about the uniform integrability.The definition I am using is that a...

ddcon4r

ddcon4r

Answered

2022-07-01

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 > 0, there exists a k such that for any x in χ we have E [ | x | I { x k } ] < 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 ?

Answer & Explanation

Ordettyreomqu

Ordettyreomqu

Expert

2022-07-02Added 22 answers

A family of random variables ( X α ) α A is u.i. if
sup α A E | X α | 1 { | X α | > M } 0
as M . Note that X α '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 | > M } E | Y α | 1 { | Y α | > M } .

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