The limit of binomial means of a sequence

Abstract

For a sequence $\{a_n\}_{n\geq 0}$ of real numbers and for a parameter $0<p<1$, we define the sequence of its arithmetic means $\{a^*_n\}_{n\geq 0}$ and the sequence of its $p$-binomial means $\{a^p_n\}_{n\geq 0}$ as \begin{align*} a^*_n=\frac{1}{n+1}\sum_{i=0}^n a_i & & \textrm{and} && a^p_n=\sum_{i=0}^n\binom{n}{i}p^i(1-p)^{n-i} a_i. \end{align*} We compare the convergence of sequences $\{a_n\}_{n\geq 0}$, $\{a_n^*\}_{n\geq 0}$ and $\{a_n^p\}_{n\geq 0}$ for various $0<p<1$, i.e. we analyze when the convergence of one sequence implies the convergence of the other. While the sequence $\{a^*_n\}_{n\geq 0}$, known also as the sequence of Ces\`{a}ro means of a sequence, is well studied in the literature, the results about $\{a^p_n\}_{n\geq 0}$ are hard to find. Our main result shows that, if $\{a_n\}_{n\geq 0}$ is a sequence of non-negative real numbers such that $\{a^p_n\}_{n\geq 0}$ converges to $a\in\mathbb{R}\cup\{\infty\}$ for some $0<p<1$, then $\{a^*_n\}_{n\geq 0}$ also converges to $a$. We give an application of this result on finite Markov chains.