Actually, I think the answer to your question is yes, such a matrix does exist in the 2x2 case. Can do something similar to the construction of the Liouville constant. I'll write up a full answer in a mo...

Yes, cos(nt) is dense in [-1,1] when theta is irrational. However, on its own that doesn't imply that 2 r^n cos(nt) either is or isn't dense in the reals. If theta is algebraic then you can use Roth's theorem to deduce that cos(nt) can't be less than (1/n^2) infinitely often, so 2r^ncos(nt)-> infinity. However, there is still the transcendental theta case.

Victor, if it has complex eigenvalues and r>1 then the traces are 2r^n cos(nt). Does this have to grow with n?

btw, point 2 can be stated quickly using M(x^(n!))>n.

btw, I'm guessing that you also post on Wilmott (or, at least, someone else asked the same question). Don't know if there's any good answers except for certain special case distributions, such as Gaussians.

I don't have any clever mathematical answer to this, just pointing out a situation where this problem arises. Given three currencies, say JPY, USD and EUR then the distributions (in a risk neutral measure) of the spots of the crosses USD/JPY, EUR/JPY, EUR/USD can each be implied from the options markets. Furthermore, log(Spot(USD/JPY))=log(Spot(EUR/JPY))-log(Spot(EUR/USD)). So, solving your problem is required to imply the joint distributions of these fx spots at any given time from the options markets, so that you can then price derivatives depending on multiple FX crosses.

Yes, those two will be zero by the zero-one law as you state. For the converse you can use the moment generating or characteristic function. Sorry, not able to give a full answer right now

I'd be interested in seeing your counterexample, if you're going to post it sometime.

Actually, I think Jonathan's definition in words above does have a misprint, and is satisfied by the trivial topology. "...should have a basis of neighborhoods which are products..." fixes it.

Isn't this just a matter of reading it off the moment generating function? Gives alpha(alpha+1)...(alpha+n-1)beta^n.

Further to my comment above, the more general example in my second edit can also be understood in terms of local times of Brownian motion. X_A is the local time at 0 of a Brownian motion B while its maximum process B*(t)=max_{s<=t}B(s) is in A.