
06505030
j
2016d.00781
Schmuland, Byron
Linear recurrences via probability.
Am. Math. Mon. 122, No. 4, 386389 (2015).
2015
Mathematical Association of America (MAA), Washington, DC
EN
I75
K65
linear mean recurrences
convergence
Markov chain
renewal theorem
Zbl 1303.39002
doi:10.4169/amer.math.monthly.122.04.386
{\it D. Borwein} et al. [Am. Math. Mon. 121, No. 6, 486498 (2014; Zbl 1303.39002)] have presented various approaches to studying the convergence of the recurrently defined sequence $$ x_n = \alpha_m x_{n1} + \alpha_{m1} x_{n2} + \cdots + \alpha_1 x_{nm} , \quad n>m, \eqno(1) $$ with initial conditions $x_n=a_n$, $n=1,\dots, m$. The author of the paper under review offers two more approaches based on probability theory provided the $\alpha_j$ are positive. Namely, Markov chain techniques allow one to represent the limit $L$ of $(x_n)$ as $L=\sum_{k=1}^m \alpha_k \pi_k$, with $\pi$ the stationary distribution of the chain that is canonically associated to (1). Another approach makes use of renewal theory. I completely agree with the author's conclusion that ``this delightful problem could also enliven an intermediate probability class.''
Dirk Werner (Berlin)