Random Walk on Z^+

Discussion in 'Probability' started by Apus Chuh, Sep 3, 2011.

  1. Apus Chuh

    Apus Chuh Guest


    Let W be a random walk on Z^+ = {0,1,2,...} with initial state 0 and
    increments of +1/0/-1. At each time, the random walk has a 'state'
    which is either on or off. The increments W'_i = W_i - W_{i-1} are
    known to satisfy: there exist a,b>0 such that

    If 'off' at time i-1, then
    P ( W'_i = +1, switches 'on' at time i ) >= a,
    P ( W'_i = -1, stays 'off' at time i ) <= 1-a.

    Else if 'on' at time i-1, then
    P ( W'_i = +1, switches 'off' at time i ) <= b,
    P ( W'_i = -1, stays 'on' at time i ) >= 1-b.

    Does there exist an exponential bound P ( W_1,...,W_r > 0 ) <= e^{-cr}
    for some c>0?

    We may assume b<1/2 as well. Then the empirical idea is that the
    random walk will switch 'on' very quickly and then by equation 4, stay
    'on' and decrease down to 0 very quickly. But I'm not sure how to
    quantify this. Any help would be greatly appreciated,

    Apus Chuh, Sep 3, 2011
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