How to create a compact subset of function space?

[Testguy]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

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

$(S_r,d)$ 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.

 

[mighty2000]

Thank you for your question. It is indeed possible to construct a compact subset from the set of all cumulative distribution functions (cdf) using a clever metric. One approach is to use the L^p metric, defined as d_p(F_1,F_2) = \left( \int_{-\infty}^\infty |F_1(x) - F_2(x)|^p dx \right)^{1/p}, where p > 0. This metric is known to generate a complete and compact metric space.

To restrict the set further, you can define S_r = \{F \in S: d_p(F,R) < r\}, where R is a pre-defined cdf and r > 0. This will give you a subset of S that is both compact and complete.

I hope this helps. Good luck with your research!



Scientist