Random walks and periodic continued fractions
Abstract
<jats:p>Nearest-neighbour random walks on the non-negative integers with transition probabilities <jats:italic>p</jats:italic><jats:sub>0,1</jats:sub> = 1, <jats:italic>p<jats:sub>k,k</jats:sub></jats:italic><jats:sub>–1</jats:sub> = <jats:italic>g<jats:sub>k</jats:sub></jats:italic>, <jats:italic>p<jats:sub>k,k</jats:sub></jats:italic><jats:sub>+1</jats:sub> = 1– <jats:italic>g<jats:sub>k</jats:sub></jats:italic> (0 < <jats:italic>g<jats:sub>k</jats:sub></jats:italic> < 1, <jats:italic>k</jats:italic> = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (<jats:italic>g<jats:sub>k</jats:sub></jats:italic>) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.</jats:p>
