Question: Random walk inside circles. Fix r > 0. Draw a circle centered at (0, 0) with radius r, Let (X₁, Y₁) be picked uniformly at random from the area inside this circle. Given (X₁, Y₁) draw a circle with radius r centered at (X₁, Y₁), Let (X2, Y2), be picked uniformly at random from the area inside this circle, and so on. Given (X₁, Y₁)  . ..  (X_n, Y_n), let (X_n+1, Y_n+1) be picked uniformly at random from the area inside the circle around (X_n, Y_n) with radius r.

a) Find r so that for large n the distribution of Xn in this problem is nearly the same as for a square of side 2 instead of the circle of radius r. [Hint Find E[X²₁] by considering E[Y²₁] as well.]

b) Are X_n and Y_n independent?

c) The point (X_n, Y_n) is projected onto the line rotated an angle θ from the X-axis at X_n cos θ + Y_n sin θ measured from the origin along this line. Use the normal approximation for sums of independent random variables to show that with r as in part a), for every θ ? [0, 2π] and for large n, the distribution of X_n Cos θ + Y_n sin θ is nearly the same for both the circle of radius r and the square of side 2.

d) It is known that a joint distribution of (X, Y) in the plane is determined by the distributions of all the projections X Cos θ + Y Sin θ as θ ranges over [0, 2π]. In particular if X and Y are independent standard normal variables. An approximate version of this result is also true: if X Cos θ + Y Sin θ has approximately the standard normal: distribution for every θ, then X and Y are approximately independent standard normal variables. Apply this result and part c) to approximate the probability that for r as in part a) and n = 300, the point (X_n, Y_n) defined using circles of radius r lies outside the circle of radius 10 centered at the origin.