PDF of random motion - similar to Browninan motion

[jaumzaum]

Hello guys, and sorry for my english in advance.

I was presented some time ago with the following problem:
Suppose there is a frog that jumps in any direction randomly, and all the jumps have size 1. What's the probability of, after 3 jumps, the frog be less than 1 unit from the origin.

I solved the problem with a double integral (if I remember well the answer is 25%), but then I thought about a similar and more general problem that I found out it's a lot similar to the Browninan motion.

Suppose there is a particle in a 2-D space that moves only in displacements of size 1 and random directions that follow a uniform probability density. What is the PDF of the particle distance from the origin after N moves? What is the probability of, after N movements, the particle end in a distance less then kN from the origin? N>>1


I approached it in the following way (and I couldn't finish):Let $\theta_i ∈ [0, 2\pi)$
Final position: $(∑cos(\theta_i), ∑sin(\theta_i))$
So $(∑cos(\theta_i))^2 + (∑sin(\theta_i))^2 < k^2N^2$
Define $\theta_{ij} = \theta_i - \theta_j ∈ (-2\pi, 2\pi)$
$∑cos(\theta_{ij})) < (k^2 N^2 - N)/2$

The PDF of $\theta_ij$ is not uniform, but if we define
$\alpha_{ij} = \begin{cases} x & \text{if } 0 \leq x < \pi \\ 2\pi - x & \text{if } \pi \leq x < 2\pi \\ \end{cases}$

Then $\alpha_{ij}$ has a uniform distribution and covers all the values of $cos(x)$, that way we can define $C_{ij} = cos(\alpha_{ij})$ with probbility density function $1/sqrt(1-x^2)$ and

$∑C_{ij} < (k^2N^2-N)/2$

but I do not find a way to calulate the PDF of the N (N-1)/2 variables above.

Can anyone help me?

 

[Orodruin]

I suggest using the central limit theorem.