What is the non-lattice definition of non-lattice?

[techmologist]

definition of "non-lattice"

In section 3 of this paper (bottom of 4th page):

http://www.bjmath.com/bjmath/breiman/breiman.pdf

THEOREM 1. If the random variables W^*₁, W^*₂, ... are nonlattice, then for any strategy ...

What does nonlattice mean? Thank you.

 

[SW VandeCarr]

What does nonlattice mean? Thank you.

A lattice can be represented by a discrete subspace which spans the vector space $$R^n$$. Any point which cannot be generated from the basis vectors by a linear combination with integer coefficients is a non-lattice point (a point with at least one irrational coordinate).

 

[techmologist]

A lattice can be represented by a discrete subspace which spans the vector space $$R^n$$. Any point which cannot be generated from the basis vectors by a linear combination with integer coefficients is a non-lattice point (a point with at least one irrational coordinate).

Yeah, that's the only mathematical notion of lattice I am familiar with. Like in crystal structures. But I wasn't sure what it meant in this context: "nonlattice random variables". Is it just a fancy way of saying that the random variables are continuous--or that they attain their limiting values or something like that?

 

[SW VandeCarr]

Yeah, that's the only mathematical notion of lattice I am familiar with. Like in crystal structures. But I wasn't sure what it meant in this context: "nonlattice random variables". Is it just a fancy way of saying that the random variables are continuous--or that they attain their limiting values or something like that?

I don't know. I've seen several papers that use this terminology instead of "continuous". Here's one:

http://econpapers.repec.org/paper/pramprapa/4120.htm

It must have something to do with the modeling of games in terms of "equilibrium sets".

 

[g_edgar]

A "lattice" random variable has all values integer multiples of some one number. This is not the same as "discrete" random variable. For example, if $X$ has only the values 1 and $\sqrt{2}$ is would be discrete but not lattice.

 

[SW VandeCarr]

A "lattice" random variable has all values integer multiples of some one number. This is not the same as "discrete" random variable. For example, if $X$ has only the values 1 and $\sqrt{2}$ is would be discrete but not lattice.

OK. So a discrete RV can be non-lattice provided it ranges over a countable set? (It's a rhetorical question. No need to respond unless you disagree.) Thanks.

 

[techmologist]

A "lattice" random variable has all values integer multiples of some one number. This is not the same as "discrete" random variable. For example, if $X$ has only the values 1 and $\sqrt{2}$ is would be discrete but not lattice.

Thank you for that definition :).

I don't know. I've seen several papers that use this terminology instead of "continuous". Here's one:

http://econpapers.repec.org/paper/pramprapa/4120.htm

It must have something to do with the modeling of games in terms of "equilibrium sets".

Thanks for the link.