id: 06505030
dt: j
an: 2016d.00781
au: Schmuland, Byron
ti: Linear recurrences via probability.
so: Am. Math. Mon. 122, No. 4, 386-389 (2015).
py: 2015
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: I75 K65
ut: linear mean recurrences; convergence; Markov chain; renewal theorem
ci: Zbl 1303.39002
li: doi:10.4169/amer.math.monthly.122.04.386
ab: {\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 $$ x_n = α_m x_{n-1} +
α_{m-1} x_{n-2} + \cdots + α_1 x_{n-m} , \quad n>m, \eqno(1) $$ 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 $α_j$ are positive. Namely, Markov chain techniques allow
one to represent the limit $L$ of $(x_n)$ as $L=\sum_{k=1}^m α_k
π_k$, with $π$ 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.”
rv: Dirk Werner (Berlin)