On convergence of binomial means, and an application to finite Markov chains (Q2827790)

For a sequence \(\{a_n\}_{n\geq0}\) of real numbers and for a parameter \(0<p<1,\) define the sequence of its arithmetic means \(\{a_n^*\}_{n\geq0}\) and the sequence of its \(p\)-binomials means \(\{a_n^\rho\}_{n\geq0}\) as NEWLINE\[NEWLINEa_n^*={1\over n+1}\sum_{i=0}^{n}a_i, \quad \text{and} \quad a_n^p=\sum_{i=0}^{n}\binom{n}{i}p^iq^{n-i}a_i,NEWLINE\]NEWLINE where \(q=1-p\).NEWLINENEWLINEIn this article, the authors analyse the relationship between the sequences \(\{a_n\}_{n\geq0}\), \(\{a_n^p\}_{n\geq0}\) and \(\{a_n^*\}_{n\geq0}\).NEWLINENEWLINETheir results are:{\parindent=0.9cm \begin{itemize}\item[(1)] \(\{a_n\}_{n\geq0}\Rightarrow\{a_n^{p_1}\}_{n\geq0}\) \item[(2)] \(\{a_n\}_{n\geq0}\Rightarrow\{a_n^{p_2}\}_{n\geq0}\) \item[(3)] \(\{a_n\}_{n\geq0}\Rightarrow\{a_n^{*}\}_{n\geq0}\) \item[(4)] \(\{a_n^{p_1}\}_{n\geq0}\) and \(a_n\geq0\Rightarrow\{a_n^{*}\}_{n\geq0}\) \item[(5)] \(\{a_n^{p_2}\}_{n\geq0}\Rightarrow\{a_n^{p_1}\}_{n\geq0}\) \item[(6)] \(\{a_n^{p_2}\}_{n\geq0}\) and \(a_n\geq0\Rightarrow\{a_n^{*}\}_{n\geq0}\) \item[(7)] \(\{a_n^{*}\}_{n\geq0}\nRightarrow\{a_n\}_{n\geq0}\) \item[(8)] \(\{a_n^{*}\}_{n\geq0}\nRightarrow\{a_n^{p_1}\}_{n\geq0}\) \item[(9)] \(\{a_n^{*}\}_{n\geq0}\nRightarrow\{a_n^{p_2}\}_{n\geq0}\) \item[(10)] \(\{a_n^{p_1}\}_{n\geq0}\nRightarrow\{a_n\}_{n\geq0}\) \item[(11)] \(\{a_n^{p_2}\}_{n\geq0}\nRightarrow\{a_n\}_{n\geq0}\) \item[(12)] \(\{a_n^{p_1}\}_{n\geq0}\) and ?\(a_n\geq0\Rightarrow\{a_n^{p_2}\}_{n\geq0}\) NEWLINENEWLINE\end{itemize}} where \(0<p_1<p_2<1.\)NEWLINENEWLINEThe symbol \(\Rightarrow\) means that the implication of convergence holds, and the symbol \(\nRightarrow\) means that there is a counterexample with \(a_n\in\{0,1\}\), for all \(n\in\mathbb{N}\). If there is a condition before \(\Rightarrow\), then the implication does not hold in general, but it holds if the condition is true. If there is a ? before the condition, the author does not know whether the condition is the right one (an open problem), but the implication does not hold in general.NEWLINENEWLINEFinally, the author gives an application of his theorem to finite Markov chains.