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Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$.

The walk terminated whenever it either steps on the right boundary at $40$ or oversteps it by $1$ (i.e, terminates at $41$). The left boundary at $0$ is reflecting: whenever it is reached, the next step is either $1$ or $3$, with probabilities $\frac{2}{9}$ and $\frac{7}{9}$, respectively.

I would like to determine the expected time until the walk terminates.

I have calculated the expected value of the random walk in this question, however I have no idea how to answer when it will terminate. Any help is appreciated.

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  • $\begingroup$ The reflection would make more sense at $1$ than at $0$ if the next positions are $1$ or $3$ $\endgroup$
    – Henry
    Commented Aug 12, 2021 at 15:57
  • $\begingroup$ I do not see a simple way of doing this, but if you adjust the reflection as I suggested then the expected number of steps from $1$ to $40$ or $41$ seems to be about $27.229$, which is not far from $\frac9{13}\left(\frac{40+41}2-1\right)$ $\endgroup$
    – Henry
    Commented Aug 12, 2021 at 16:22

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