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Let the following experience :

Let $x=n_0\in\mathbb{N}$.

Each step, $x$ has a $1/2$ probability of increasing by one and a $1/2$ probability of decreasing by one.

The goal is to determine the Mean of this experience for an infinite number of steps.

If $x$ was varying over $\mathbb{Z}$ then it seems obvious to me that the Mean should be $n_0$. However, if we now limit $x$ to $\mathbb{N}$ (which implies that if $x$ is $0$ and is to decrease then it stays at $0$), what is the Mean of the experience ?

SBF
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Hippalectryon
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1 Answers1

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When you have equal probability of moving +1 or -1 then there will be no stable distribution and so the mean is not defined.

See here.

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lemon
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