# Definition of non-lattice

definition of "non-lattice"

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

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

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

What does nonlattice mean? Thank you.

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).

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?

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".

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.

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.

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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 :).

SW Vandecarr said:
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".