[HP]

I'm still uneasy about this all however

I had the money markets in mind when I made the OP

Let's assume an efficient market (random walk)

And let's simplify stuff and say the interest rates in the USA and Europe will always be the same. If you don't know why this matters don't worry

so, With similar logic to the above you could say that looking at the USD vs the EUR, through a random walk the USA will approach 0

But then the same could be said for the EUR vs the USD

I'm still pondering about what kind of probability density functions we have for the change of USD/EUR after a day

hmm...