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Department of Mathematical Sciences

Seminar Archives

On this page you can find information about seminars in this and previous academic years, where available on the database.

Statistics Seminars: On the centre of mass of random walks

Presented by Hugo Lo, Durham University

24 April 2017 14:40 in CM221

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.

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