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This is E10.10 From Williams "Probability with Martingales" book, which I spent a lot but I could not figure out what is the relation between Gauss's theorem and this end! Here is the question:

The control system on the star-ship Enterprise has gone wonky. All that one can do is to set a distance to be travelled. The spaceship will then move that distance in a randomly chosen direction, then stop. The object is to get into the Solar System, a ball of radius r. Initially, the Enterprise is at a distance $R_o(> r)$ from the Sun. Let $R_n$ be the distance from Sun to Enterprise after n 'space-hops'. Use Gauss's theorems on potentials due to spherically-symmetric charge distributions to show that whatever strategy is adopted, $1/ R_n$ is a supermartingale, and that for any strategy which always sets a distance no greater than that from Sun to Enterprise, $1/R_n$ is a martingale.

How could it be related to something like $1/R_n$ in any way? What is a good idea?

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It's known from potential theory that a spherically symmetrical charge distribution generates the same potential outside it as if the total charge were concentrated at the centre. The expected value of $1/R_{n+1}$ is the potential generated at the sun by a uniformly charged spherical surface around the Enterprise's position $X_n$ at step $n$. If the distance set is no greater than that from the sun to the Enterprise, the sun is outside this surface, so the potential is the same as that generated by the Enterprise at $X_n$, that is, $1/R_n$, so $1/R_n$ is a martingale.

On the other hand, the potential inside a uniformly charged spherical surface is constant; that is, it's equal to its value at the surface, which is the potential the total charge concentrated at the centre would generate at the surface, and thus less than the potential the total charge concentrated at the centre would generate inside the sphere. Thus, if the distance set may be greater than that from the sun to the Enterprise, the sun may be inside the uniformly charged spherical surface, and the expected value of $1/R_{n+1}$ may be less than $1/R_n$, so $1/R_n$ is a supermartingale.

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