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

David Gajser

Ars Mathematica Contemporanea, Tome 10 (2016) no. 2, pp. 393-410.

Résumé

For a sequence {an}n ≥ 0 of real numbers, we define the sequence of its arithmetic means {an * }n ≥ 0 as the sequence of averages of the first n elements of {an}n ≥ 0. For a parameter 0  p  1, we define the sequence of p-binomial means {anp}n ≥ 0 of the sequence {an}n ≥ 0 as the sequence of p-binomially weighted averages of the first n elements of {an}n ≥ 0. We compare the convergence of sequences {an}n ≥ 0, {an * }n ≥ 0 and {anp}n ≥ 0 for various 0  p  1, , we analyze when the convergence of one sequence implies the convergence of the other.While the sequence {an * }n ≥ 0, known also as the sequence of Cesàro means of a sequence, is well studied in the literature, the results about {anp}n ≥ 0 are hard to find. Our main result shows that, if {an}n ≥ 0 is a sequence of non-negative real numbers such that {anp}n ≥ 0 converges to a ∈ R ∪ {∞} for some 0  p  1, then {an * }n ≥ 0 also converges to a. We give an application of this result to finite Markov chains.

Détail

DOI : 10.26493/1855-3974.705.56d

Keywords: Sequence, convergence, Cesaro mean, binomial mean, finite Markov chain