# Definition of non-lattice

1. Aug 4, 2010

### techmologist

definition of "non-lattice"

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

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

What does nonlattice mean? Thank you.

2. Aug 5, 2010

### SW VandeCarr

Re: definition of "non-lattice"

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).

3. Aug 5, 2010

### techmologist

Re: definition of "non-lattice"

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?

4. Aug 5, 2010

### SW VandeCarr

Re: definition of "non-lattice"

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".

5. Aug 5, 2010

### g_edgar

Re: definition of "non-lattice"

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.

6. Aug 5, 2010

### SW VandeCarr

Re: definition of "non-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.

Last edited: Aug 5, 2010
7. Aug 5, 2010

### techmologist

Re: definition of "non-lattice"

Thank you for that definition :).