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    • Thread Starter

    I'm stuck with this question about random walks on Z^2

    Let S_n = (X_n, Y_n) be a simple symmetric random walk in Z^2, starting from (0, 0), and set T = inf {n >= 0: max{|X_n|, |Y_n|} = 2}.Calculate E(T) (Expectation of T).

    Thanks in advance

    Michelle (Mic)
    • Thread Starter


    There aren't actually many possibilities: there are only 4 states you need to consider

    A: particle is at the origin
    B: particle is distance 1 from the origin
    C: particle is distance sqrt(2) from the origin.
    D: particle is >=2 from the origin (so max|X_n|,|Y_n| = 2).

    Write T_B for inf {n >= 0: max{|X_n|, |Y_n|} = 2} when you start from state B (and T_C, T_D similarly. Clearly T_D = 0).

    If you think about what happens if you take one "step" from state A, it's obvious that T = T_B + 1.
    Now think about what happens if you take one "step" from state B. What's the chance you end up in state A? State C? State D? So we can write a linear equation relating T, T_B, T_C.
    Do the same for state C.

    So, 3 equations, 3 unknowns. Solve for T, T_B, T_C.
    • Thread Starter

    ok, I understand this, but where does this lead to? Having known these, how would i compute E(T)?
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Updated: February 6, 2010
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