Statistical limit superior and limit inferior

Fridy, J.; Orhan, C.

doi:10.1090/S0002-9939-97-04000-8

Statistical limit superior and limit inferior
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by J. A. Fridy and C. Orhan

Proc. Amer. Math. Soc. 125 (1997), 3625-3631

DOI: https://doi.org/10.1090/S0002-9939-97-04000-8

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Abstract:

Following the concept of statistical convergence and statistical cluster points of a sequence $x$, we give a definition of statistical limit superior and inferior which yields natural relationships among these ideas: e.g., $x$ is statistically convergent if and only if $\textrm {st}\text {-}\textrm {liminf} x= \textrm {st}\text {-}\textrm {limsup} x$. The statistical core of $x$ is also introduced, for which an analogue of Knopp’s Core Theorem is proved. Also, it is proved that a bounded sequence that is $C_{1}$-summable to its statistical limit superior is statistically convergent.

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