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

Abstract

For a sequence {a_n}_n ≥ 0 of real numbers, we define the sequence of its arithmetic means {a_n^*}_n ≥ 0 as the sequence of averages of the first n elements of {a_n}_n ≥ 0. For a parameter 0 < p < 1, we define the sequence of p-binomial means {a_n^p}_n ≥ 0 of the sequence {a_n}_n ≥ 0 as the sequence of p-binomially weighted averages of the first n elements of {a_n}_n ≥ 0. We compare the convergence of sequences {a_n}_n ≥ 0, {a_n^*}_n ≥ 0 and {a_n^p}_n ≥ 0 for various 0 < p < 1, , we analyze when the convergence of one sequence implies the convergence of the other.

While the sequence {a_n^*}_n ≥ 0, known also as the sequence of Cesàro means of a sequence, is well studied in the literature, the results about {a_n^p}_n ≥ 0 are hard to find. Our main result shows that, if {a_n}_n ≥ 0 is a sequence of non-negative real numbers such that {a_n^p}_n ≥ 0 converges to a ∈ R ∪ {∞} for some 0 < p < 1, then {a_n^*}_n ≥ 0 also converges to a. We give an application of this result to finite Markov chains.