[jason1990]

I agree that the initial conditions in the problem are ambiguous. I interpreted the statement "hopping on the integers from minus infinity to plus infinity" as "hopping on the integers, with no a priori restriction as to how negative or how positive his location can become."

How would we rigorously define a stochastic process which models this frog's movements and which starts at -infinity? We could try to take this literally, and add a point at infinity to the integers. But then, what is the distribution of the frog's first jump? He must eventually make a first jump to a finite integer. After that initial jump, there will be a lowest integer below which all integers are missed.

Or we could try to take some limiting procedure, by considering a sequence of processes where he starts at x, and let x go to -infinity. But again, for each fixed process in the sequence, there will be a lowest integer below which all integers are missed. If his starting point is any fixed integer, or even any integer-valued random variable, then the answer will be 1 - M. So I cannot see how we could take a limit in any reasonable way and get Y.

In fact, this reminds me a bit of the martingale betting system. Any reasonable limiting procedure shows that it is a disaster, whereas naively "starting at infinity" seems to show that it is a goldmine.