A Few Random Walk Questions

[HP]

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If you take those values you will get a random walk that will reach every point [..., 0.68, 0.75, 0.83, 0.91, 1, 1.1, 1.21, 1.33, ...] infinitely often.

You might want to google 'geometric random walk' for some more infos.

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Yes, this was my intuitive guess for perfectly efficient markets (or else it would be weird, as pointed out in my above post)

which leads me to another problem (posted in business forum, and what led to this OP):

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Okay I had a highly theoretical, probably not practical at all, thought the other day. Feel free to ignore if it's pointless to discuss

Assume:

a perfectly efficient money market
it is possible to borrow and lend cash at the same interest rate, in the US and in Europe
there will always be a minimum amount variance in the USD/EUR
the USD and EUR will always exist

Okay so basically we buy 100 USD worth of EUR today, and put the EUR in the bank earning interest.

We buy back USD once the exchange rate is such that we make ten times as much USD than we would have if we had just put it into a US bank account to begin with

Now, if I've understood random walk theory correctly, if there will always be a minimum amount of variance, and it's always a random walk, the chance the exchange rate will eventually hit our target (so we can make ten times as much) is 100%. It may take a number generations and our decedents may be the only ones profiting, but we don't mind we are nice people

This seems counterintuitive - it's like we are beating the perfectly efficient market. Thoughts?

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[HP]

for starters, if you don't understand why the difference in interest rates between the USA and Europe matters, don't worry just ignore it and assume they are always the same