@article {MATHEDUC.06505030,
author = {Schmuland, Byron},
title = {Linear recurrences via probability.},
year = {2015},
journal = {American Mathematical Monthly},
volume = {122},
number = {4},
issn = {0002-9890},
pages = {386-389},
publisher = {Mathematical Association of America (MAA), Washington, DC},
doi = {10.4169/amer.math.monthly.122.04.386},
abstract = {{\it D. Borwein} et al. [Am. Math. Mon. 121, No. 6, 486--498 (2014; Zbl 1303.39002)] have presented various approaches to studying the convergence of the recurrently defined sequence $$ x_n = \alpha_m x_{n-1} + \alpha_{m-1} x_{n-2} + \cdots + \alpha_1 x_{n-m} , \quad n>m, \eqno(1) $$ 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.''},
reviewer = {Dirk Werner (Berlin)},
msc2010 = {I75xx (K65xx)},
identifier = {2016d.00781},
}