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How to create a compact subset of function space?

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Let S be the set of all cumulative distribution functions (cdf), as defined in basic statistics (defined below).

    Along with a metric d, (S,d) will defined a metric space.

    My goal is (with any metric) to make a compact subset from this set S. Does anyone have any idea on how I can possibly do that? Is there any "clever" metric that I can use to do this, or is maybe not possible? Any advice is very welcome.

    This is really not homework, but to me the question seemed to fit better in here.


    2. Relevant equations

    A cdf is defined as the set of all bounded, right-continuous, non-decreasing functions from the reals to [0,1] with [itex]\lim_{x \rightarrow \infty} F(x)=1[/itex] and [itex]\lim_{x \rightarrow -\infty} F(x)=0[/itex].

    3. The attempt at a solution

    I have so far used the supremum (or is it called uniform?) metric [itex]d(F_1,F_2) = \sup_x |F_1(x) - F_2(x)|[/itex] to create a metric space, and restricted the set by defining [itex] S_r= \{F \in S: d(F,R)<r \}[/itex], where} [itex]R[/itex] is some pre-defined cdf.

    [itex](S_r,d)[/itex] does however not seem to establish a compact space. I have shown that this space is complete, but it does not seem like this set is totally bounded - at least all my attempts on proving that it is has lead to counterexamples.

    Thank you for any advice.
     
  2. jcsd
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