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