Not an answer, but I was exploring a similar random walk, so I thought I would include an image of a simulation. In my walk, each step is of a random length drawn from a normal distribution with mean $\mu=0$ and $\sigma=1$. So $x_{i+1} = x_i + \cal{N}$$(0,1)$ if that is nonnegative, and otherwise $x_{i+1}=0$.

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            ![RandWalk0][1]

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The walk wanders rather far from zero.

Not an answer, but I was exploring a similar random walk, so I thought I would include an image of a simulation. In my walk, each step is of a random length drawn from a normal distribution with mean $\mu=0$ and $\sigma=1$. So $x_{i+1} = x_i + \cal{N}$$(0,1)$ if that is nonnegative, and otherwise $x_{i+1}=0$.

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            ![RandWalk0][1]

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Vertical axis is $x_i$; horizontal $i$, the number of steps. The walk wanders rather far from zero.

Not an answer, but I was exploring a similar random walk, so I thought I would include an image of a simulation. In my walk, each step is of a random length drawn from a normal distribution with $\sigma=1$, and mean centered on the current position $x_i$. So $x_{i+1} = x_i + \cal{N}$$(x_i,1)$ if that is nonnegative, and otherwise $x_{i+1}=0$.

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            ![RandWalk0][1]

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Not an answer, but I was exploring a similar random walk, so I thought I would include an image of a simulation. In my walk, each step is of a random length drawn from a normal distribution with mean $\mu=0$ and $\sigma=1$. So $x_{i+1} = x_i + \cal{N}$$(0,1)$ if that is nonnegative, and otherwise $x_{i+1}=0$.

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            ![RandWalk0][1]

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The walk wanders rather far from zero.