Timeline for When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 10 at 6:21 | answer | added | user1536 | timeline score: 1 | |
| May 5 at 17:14 | history | edited | Michael Hardy | CC BY-SA 4.0 |
deleted 4 characters in body
|
| May 3 at 18:53 | comment | added | Roger Van Peski | Thanks everyone, I've added more detail about the setting I'm after (I didn't put it in earlier because I figured there's probably a general theorem and didn't want to cloud the question with details, but it looks like maybe this isn't the case). @JulianHölz I think in my example one can get away with sequences rather than nets, though in more general settings I agree you're right. | |
| May 3 at 18:51 | history | edited | Roger Van Peski | CC BY-SA 4.0 |
added 932 characters in body
|
| May 3 at 17:58 | comment | added | Julian Hölz | By cadlag I mean that for every increasing net $(x_\lambda)$ converging to $x$ we have that the limit of $f_\mu(x_\lambda)$ exists and for decreasing nets $(x_\lambda)$ with limit $x$ we have $f_\mu(x_\lambda) \to f_\mu(c)$. Sequences will probably not suffice in general. | |
| May 3 at 17:57 | comment | added | Sam Hopkins | I think you probably want some further finiteness conditions on the posets you're considering, Roger. For example, the set of rational numbers $\mathbb{Q}$ with its usual total order is a countable lattice where each element covers finitely many elements (in fact, no elements at all!) but is probably not what you had in mind. A possible additional condition you might want is "locally chain finite," i.e., no interval $[x,y]$ contains an infinite chain (although it may be infinite). | |
| May 3 at 17:56 | comment | added | Iosif Pinelis | @SamHopkins : Thank you for your comment. | |
| May 3 at 17:55 | comment | added | Sam Hopkins | @IosifPinelis for $x,y \in P$, we say $y$ covers $x$ (often written $x \lessdot y$) if $x < y$ and there is no $z\in P$ with $x < z < y$. This is standard terminology in poset theory. | |
| May 3 at 17:53 | comment | added | Iosif Pinelis | I don't think that there are any interesting non-linear orders for which these conditions will suffice. Consider e.g. the set $X$ being the set of all integral points in $[0,n]^2$ and $f(x,y)=\max(x,y)$ for $(x,y)\in X$. | |
| May 3 at 17:53 | comment | added | Julian Hölz | In the real case the function should additionally be cadlag in the order topology. You might try this condition. | |
| May 3 at 17:50 | comment | added | Iosif Pinelis | What do you mean by "covers"? | |
| May 3 at 17:44 | history | asked | Roger Van Peski | CC BY-SA 4.0 |