3-D random walk: average distance after N steps I am calculating the average distance in a 3-D random walk process after N steps. Each step is one unit long and the angle is randomly distributed around the origin. After N steps, what is the average distance from the origin?
The $X,Y,Z$ coordinates are determined by $Z=\cos(a),$ $X=\sin(a)\cos(b),$ and $Y=\sin(a)\sin(b).$ The angles $a$ and $b$ are distributed uniformly on $[0, 2\pi).$
I have simulated the process using 50,000 points. After 1 step, the average distance is 1. After 2 steps, the average distance is around 1.32. After 3 steps, the average distance is around 1.62.
How could I calculate the equation showing the average distance from the origin after N steps?
 A: Let's solve this in all dimensions $d=1,2,3,\ldots.$
The (vector) increments of the walk are $\mathbf{X}_i = (x_{1i}, x_{2i}, \ldots, x_{di}).$  After $n$ such independent steps the walk has reached the point $\mathbf{S}_n = \mathbf{X}_1 + \mathbf{X}_2 + \cdots + \mathbf{X}_n$ with corresponding components $s_{1n}, \ldots, s_{dn}.$  The question asks for the expectation of $|\mathbf{S}_n| = \sqrt{s_{1n}^2 + \cdots + s_{dn}^2}$ for large $n.$
Because the $\mathbf{X}_i$ are uniformly distributed on the unit sphere,

*

*Their components are identically distributed.  (Thus, in particular, they have identical means, variances, and covariances.  Details are given at https://stats.stackexchange.com/a/85977/919, but this additional information is not necessary for the following analysis.)


*Their means are all zero (since the spherical symmetry implies the means equal their own negatives and the boundedness of the vectors implies the means exist and are finite.)


*The variances of each $x_{ki}$ are all $1/d,$ because for any fixed $i,$ the sum of the variances of the $x_{ki}$ is the expectation of the sum of their squares, which is constantly $x_{1i}^2 + \cdots + x_{di}^2 = 1.$


*Their covariances are all zero.  This is a bit of a surprise, because the sum-to-square restriction implies the components of any $\mathbf{X}_i$ are not independent.  Nevertheless, the spherical symmetry of the distribution of $\mathbf{X}_i$ implies the distribution of $y_i=(x_{1i} + x_{2i} + \cdots + x_{di})/\sqrt{d}$ is identical to that of any of the components, whence
$$\frac{1}{d} = \operatorname{Var}\left(y_i\right) = \frac{1}{d}\sum_{j,k=1}^d E\left[x_{ji}x_{ki}\right] = \frac{1}{d}\left(d\operatorname{Var}(x_{1i}) + d(d-1)\operatorname{Cov}(x_{1i}, x_{2i})\right).$$
Upon plugging in $1/d$ for the variance term on the right, we see the last term $d(d-1)\operatorname{Cov}(x_{1i},x_{2i})$ must be zero.  Since either there is no covariance (for $d=1$) or else $d\gt 1,$ the covariance is zero, QED.
Because the increments are independent, the multivariate Central Limit Theorem (CLT) tells us the distribution of $\mathbf{S}_n$ is approximately multivariate Normal.  The approximating Normal distribution's parameters are determined by the means and variances of the $\mathbf{X}_i:$ it will have zero mean, variances of $n/d,$ and zero covariances.  Ergo,

the variables $(d/n)s_{kn}^2$ must be distributed approximately like squares of standard Normal variates and (therefore) their sum $(d/n)|\mathbf{S}_n|^2$ must be distributed approximately like the sum of squares of $d$ uncorrelated (whence independent) standard Normal variates.

By definition, a sum of independent standard Normal variables has a chi-squared distribution with $d$ degrees of freedom.  Also by definition, its square root has a chi distribution with $d$ d.f.  Its expectation is

$$ E\left[|\mathbf{S}_n|\right] = \sqrt{\frac{n}{d}}E\left[\sqrt{(d/n)s_{1n}^2 + (d/n)s_{2n}^2 + \cdots + (d/n)s_{dn}^2}\right] = \frac{\sqrt{2n}\,\Gamma((d+1)/2)}{\sqrt{d}\,\Gamma(d/2)}.$$

As a special case, when $d=2$ the right hand side is
$$\frac{\sqrt{2n}\,\Gamma((2+1)/2)}{\sqrt{2}\,\Gamma(2/2)} = \frac{\sqrt{n\pi}}{2},$$
exactly as noted in a comment to the question.  When $d=3$ (the case of the question), the right hand side is
$$\frac{\sqrt{2n}\,\Gamma((3+1)/2)}{\sqrt{3}\,\Gamma(3/2)} = \frac{2\sqrt{2n}}{\sqrt{3\pi}}.$$
To illustrate the general formula, here is a plot of $\sqrt{2n}\,\Gamma((d+1)/2) / (\sqrt{d}\,\Gamma(d/2))$ for $d=3$ (in red) along with the means of 1,000,000 simulated random walks at times $1$ through $n=30.$  They look to be in good agreement, especially for $n\gt 1.$  The differences between the means and this formula approach zero at a rate of $O(n^{-1/2})$ (plotted in blue) or better, as predicted by the CLT.

