Saturday, 25 August 2012

pr.probability - Is the infimum of the Ky Fan metric achieved?

Consider the probability space (Omega,calB,lambda) where
Omega=(0,1), calB is the Borel sets, and lambda is Lebesgue measure.



For random variables W,Z on this space, we define the Ky Fan metric by



alpha(W,Z)=inflbraceepsilon>0:lambda(|WZ|geqepsilon)leqepsilonrbrace.



Convergence in this metric coincides with convergence in probability.



Fix the random variable X(omega)=omega, so the law of X is Lebesgue measure,
that is, calL(X)=lambda.




Question: For any probability measure mu on mathbbR, does there exist
a random variable Y on (Omega,calB,lambda) with law mu so that
alpha(X,Y)=inflbracealpha(X,Z):calL(Z)=murbrace ?




Notes:



  1. By Lemma 3.2 of Cortissoz,
    the infimum above is dP(lambda,mu):
    the Lévy-Prohorov distance between the two laws.


  2. The infimum is achieved if we allowed to choose both random variables.
    That is, there exist X1 and Y1 on (Omega,calB,lambda)
    with calL(X1)=lambda, calL(Y1)=mu, and
    alpha(X1,Y1)=dP(lambda,mu).
    But in my problem, I want to fix the random variable X.


  3. Why the result may be true: the
    space L0(Omega,calB,lambda) is huge. There
    are lots of random variables with law mu. I can't think of any
    obstruction to finding such a random variable.


  4. Why the result may be false: the
    space L0(Omega,calB,lambda) is huge. A compactness
    argument seems hopeless to me. I can't think of any
    construction for finding such a random variable.


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