Since probability is quite far away from my daily buisiness, please forgive me if my use of terminology is wrong or the question is too trivial. However, I was not able to find the right keyword to find an answer by googling... I am even not sure if "random walk" is the right name for what I am going to describe.

Consider a particle which is moving around randomly in $\mathbb{R}^2$ in steps such that in every step its movement is desribed by a draw of a 2D Gaussian distribution with variance $\sigma$. In other words: From position $x_k$ at time $k$ it moves to position $x_{k+1} = x_k + d_k$ where $d_k$ is normally distributed with variance $\sigma$. If the particle starts at time $0$ at $0$, then the distribution of its position at time $N$ is Gaussian with variance $\sqrt{N}\sigma$, since this is just the addition of $N$ Gaussian random variables which amounts to the $N$-fold convolution of the Gaussian with variance $\sigma$. Am I right on this?

But my question is this: What is the distribution of $\max\{\|x_j - x_k\|\ |\ 1\leq j,k\leq N\}$ and how to you calculate it?

Finally: What is the answer to the same question if unit steps in random directions are taken, i.e. $d_k$ is uniformly distributed on the unit circle?

Pointers to literature are also appreciated.

1) W. Feller (1951), The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Stat, vol. 22, no. 3, 427-432 (link: projecteuclid.org/euclid.aoms/1177729589), (2) E. Tanre & P. Vallois, Range of Brownian motion with drift (link: www-sop.inria.fr/members/Etienne.Tanre…), (3) H. He, et al., Double lookbacks (link: som.yale.edu/~hh78/lb_9612.pdf) and (4) M. Magdon-Ismael, et al, On the maximum drawdown of a Brownian motion, (link: jstor.org/pss/3215821). $\endgroup$ – cardinal Oct 14 '11 at 1:53