### 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.

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DOI: https://doi.org/10.26493/1855-3974.705.56d

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications