The Hanover Collaboration

Whoa, double banger this week. I just felt bad only doing probability. But Joran and I seem to like them. So here’s the probability one.

Lets say I have this black box random function (or process as some like to say). I push the button on top, there is a painful grinding noise and then a number (on the reals) pops out.

If the number is greater than any previous number, I get excited and write it down.

If I press the button N times in a row without getting a new highest number, I get bored and quit.

0) On average, how many times, t, do I press the button before quitting?
1) what is P(quit|t)
2) what if instead of spitting out a number on R, it spits out a vector on R^m and I write down the max for each individual component. That is, I don’t quit unless I see N consecutive vectors none of which have a component that breaks the record for it’s position.
2b) What if rather than quitting completely if none of them improve, I quit looking at a component if that component fails to improve N times in a row. How long until I quit looking at all the components?

Feel free to assume N large if it helps the math.