The Student Puzzle Corner contains one or two problems in statistics or probability. Sometimes, solving the problems may require a literature search. Current student members of the IMS are invited to submit solutions electronically (to bulletin@imstat.org with subject “Student Puzzle Corner”).

The deadline has now been extended to June 18, 2015.

 The names and affiliations of (up to) the first 10 student members to submit correct solutions, and the answer to the problem, will be published in the next issue of the Bulletin.
The Editor’s decision is final.

Student Puzzle Corner 9

It is the turn of a probability problem this time. We are going to look at some questions about how random walks evolve over time. Random walks provide a great deal of intuition about randomness in general. Add to that the fascinating results that give the subject a great deal of structure and universality, and the variety of random phenomena that are modeled using random walks in some form or the other. Here is the exact problem of this issue. It is of a classic nature; but to state it clearly, we will first need a few definitions.

For given $d\ge 1$, suppose ${\mathbf{X}}_{\mathbf{1}}\mathbf{,}{\mathbf{X}}_{\mathbf{2}}\mathbf{,}\cdots$ are iid $d$-dimensional random vectors with common distribution $F$. Define $S_0=\mathbf{0}$ and for $n\ge 1,S_n={\mathbf{X}}_{\mathbf{1}}\mathbf{+}{\mathbf{X}}_{\mathbf{2}}\mathbf{+}\cdots {\mathbf{X}}_{\mathbf{n}}$. Then, $S_n$ is called a random walk driven by $F$. Take any fixed point $\mathbf{x}\in {\mathcal{R}}^{\mathbf{d}}$ and any fixed $ϵ>0$. The point $\mathbf{x}$ is called a recurrence point of the random walk $S_n$ if $P(||S_n–\mathbf{x}\mathbf{|}\mathbf{|}\mathbf{<}ϵ$ for infinitely many $\mathbf{n}\mathbf{)}\mathbf{=}\mathbf{1}$; i.e., $S_n$ returns to any given neighborhood of $x$, however small, infinitely many times with probability one. The set of all recurrence points $\mathbf{x}$ of the random walk $S_n$ is called the recurrent class of $S_n$.

These are all the definitions we need to state our exact problem. Here it is.

Explicitly characterize the recurrent class of $S_n$ in the following three cases:

(a) $d=1,F$ = the two point distribution with $P(X_i=\pm 1)=\frac{1}{2}$;

(b) $d=2,F$ = the uniform distribution inside the unit two dimensional ball;

(c) $d=3,F$ = the trivariate standard normal, i.e., normal with mean vector zero and the identity covariance matrix.