Consider a one dimensional random walk, in which the probability of moving left along a line is $q=1/2$ and the probability of moving right is $p=1/2$. The square root error $\langle d\_N \rangle$, which is the expectation of the absolute distance traveled after $N$ steps is known to be $\sqrt{2N/\pi}.$ I am interested in finding the square root error of a modified version of this problem, in which there are three distinct possibilities. You can move to the left, you can move to the right, or you can stay put on a given turn. There will now be three probabilities, the probability $p$ of moving left, the probability $q$ of moving right, and the probability $r$ of not moving at all. The particular problem of interest to me is when $p=q$. So we will have that $p+q+r=1$. Has this problem been studied, and what would be the reference; what would be the solution?



A question about the square root error of one dimensional random walks
Because of the central-limit-theorem, for large $N$ the absolute distance $d\_N$ converges in distribution as
$$P\_N(d\_N/\sqrt N)\to p(|X|),$$
where $X$ is a Gaussian random variable with mean zero and variance $2p=1-r$. Since $\mathbb{E}(|X|)=2\sqrt{p/\pi}$ we conclude that
$$\mathbb{E}(d\_N)\to \sqrt\frac{2(1-r)N}{\pi}.$$
So this "lazy" random walk differs from the simple random walk by a rescaling of the number of steps by a factor $1-r$.


