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(a) A particle moves randomly on a plane according to the following rules. It moves one unit along a randomly chosen direction ‰1, then it chooses a new direction ‰2 and moves one unit along that new direction, and so on. We suppose that the angles ‰i are independent and uniformly distributed on OE0; 2

. Let Dn be the distance between the starting point and the location of the particle after the nth step. Calculate the mean square displacement E.D2 n/.

(b) Suppose at the centre of a large plane there are at time zero 30 particles, which move independently according to the rules set out in (a). For every step they take, the particles need one time unit. Determine, for every n  1, the smallest number rn > 0 with the following property: With probability at least 0:9 there are, at time n, more than 15 particles in a circle of radius rn around the centre of the plane. Hint: Determine first some p0 > 1=2 such that P.

P30 iD1 Zi > 15/  0:9 for any Bernoulli sequence Z1; : : : ; Z30 with parameter p  p0.