We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Department of Mathematical Sciences

# Seminar Archives

Many random processes arising in applications exhibit a range of possible behaviours depending upon the values of certain key factors. Investigating critical behaviour for such systems leads to interesting and challenging mathematics. Much progress has been made over the years using a variety of techniques. This presentation will give a brief introduction to the asymptotic behaviour of the centre of mass of a $d$-dimensional random walk $S_n$, which is defined by $G_n=n^{−1} \sum_{i=1}^{n} S_i$, $n \ge 1$. By considering the local central limit theorem, we investigate the almost-sure asymptotic behaviour of the centre of mass process. We obtain a recurrence result in one dimension under minor assumptions; in the case of simple symmetric random walk the fact that $G_n$ returns infinitely often to a neighbourhood of the origin is due to Grill in 1988. We also obtain the transience result for dimensions greater than one. In particular, we give a diffusive rate of escape; again in the case of simple symmetric random walk the result is due to Grill.