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Let $X$ be a random walk on $\mathbb{Z}_{\ge 0}$ starting at $0$, with step size $1$, and there is a barrier at $0$ so that if one tries to move to $-1$ it stays at $0$ (non-reflecting). If we fix the number of steps, I think there is a way to calculate the expectation of the position in this case. What I am interested is: what if we modify so that if $X$ increases $3$ times in a row, then the step size increases to $2$ until $X$ decreases. Can we calculate the expectation of the position in this case? It seems to be close to $N/8$ where $N$ is the number of steps, but the barrier condition makes me struggle.

Thank you!

Peter Woolfitt
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user109870
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  • You can find an exact result, but this is going to be heavy as you need to condition on the 3 steps before and on the fact that it is positive. The variable you're looking at is $Y_{n+1}=max(0,Y_n + \varepsilon_{n+1}(1+\mathbb{1}{(\varepsilon_{n}>0) \times \mathbb{1}{(\varepsilon_{n-1}>0) \times \mathbb{1}{(\varepsilon_{n-2}>0))$. Now condition on $Y_n \geq 2, =1,=0$ and $\varepsilon_{n+1},\varepsilon_{n},\varepsilon_{n-1},\varepsilon_{n-2}$. – Matt B. Jul 21 '14 at 17:38
  • How to find an exact result from it? Thank you. – user109870 Jul 21 '14 at 17:54

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