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opapa
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Consider a one-dimensional random walk on the integer lattice, starting at 0. The next step is decided to be +1 or -1 with equal probability 0.5. The hitting time (the expected time required for the random walk to reach any integer α -- also called the first-passage time) will be [itex]\Theta(\alpha^2)[/itex], since random walks are martingales. Similarly, given two numbers (α<0 and β>0), the expected time for the random walk to reach either α or β is α*β.
In my case, I have two or more independent random walks r1, r2, …, that run in parallel, and I want to find the expected time required for at least one of these random walks ri to reach either αi or βi (notice that the boundaries αi and βi can be different for each random walk). My initial intuition was that, since these random walks are independent, I could find the hitting time for each random walk and get the minimum, but I found this to be wrong through a computer simulation.
Any pointers?
Thanks
In my case, I have two or more independent random walks r1, r2, …, that run in parallel, and I want to find the expected time required for at least one of these random walks ri to reach either αi or βi (notice that the boundaries αi and βi can be different for each random walk). My initial intuition was that, since these random walks are independent, I could find the hitting time for each random walk and get the minimum, but I found this to be wrong through a computer simulation.
Any pointers?
Thanks