On convergence of binomial means, and an application to finite Markov chains - Gajser, David (Ars mathematica contemporanea, 2016)
For a sequence ?$\{a_n\}_{n \ge 0}$? of real numbers, we define the sequence of its arithmetic means ?$\{a_n^*\}_{n \ge 0}$? as the sequence of averages of the first ?$n$? elements of ?$\{a_n\}_{n \ge 0}$?. For a parameter ?$0 < p < 1$?, we define the sequence of ?$p$?-binomial means ?$\{a_n^p\}_{n \ge 0}$? of the sequence ?$\{a_n\}_{n \ge 0}$? as the sequence of ?$p$?-binomially weighted averages of the first ?$n$? elements of ?$\{a_n\}_{n \ge 0}$?. We compare the convergence of sequences ?$\{a_n\}_{n \ge 0}$?, ?$\{a_n^*\}_{n \ge 0}$? and ?$\{a_n^p\}_{n \ge 0}$? for various ?$0 < p < 1$?, i.e., we analyze when the convergence of one sequence implies the convergence of the other. While the sequence ?$\{a_n^*\}_{n \ge 0}$?, known also as the sequence of Cesaro means of a sequence, is well studied in the literature, the results about ?$\{a_n^p\}_{n \ge 0}$? are hard to find. Our main result shows that, if ?$\{a_n\}_{n \ge 0}$? is a sequence of non-negative real numbers such that ?$\{a_n^p\}_{n \ge 0}$? converges to ?$a \in \mathbb{R} \cup \{∞\}$? for some ?$0 < p < 1$, then $\{a_n^p\}_{n \ge 0}$? also converges to ?$a$?. We give an application of this result to finite Markov chains.
Keywords: binomial mean; binomska sredina; Cesaro mean; Cesarova sredina; convergence; finite Markov chain; končna Markovska veriga; konvergenca; sequence; zaporedje