A continuous random walk of length 1

Question

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity?
For one step, the distribution along the x-axis is $1/(N\pi\sqrt{1-x^2})$, I wanted to take the convolution of $N$ such functions and then take the limit but the integrals seemed hopeless, is there an easy way to find the distribution? If not, I am happy with just the answer.

Answer 1

Perhaps somewhat surprisingly, the limiting distribution is the one where the distance is certain to be $0$. As GEdgar pointed out, you can apply the central limit theorem. Along either axis, you're taking the average of $N$ identically distributed steps, and by the central limit theorem the resulting marginal distribution along either axis becomes approximated by a Gaussian with variance proportional to $1/N$. Thus the variance goes to zero and the shape becomes Gaussian, so the marginal distribution along either axis converges to a delta distribution, and thus so does the two-dimensional distribution, and hence also that of the distance.