Statistical limit superior and limit inferior

Source abstract

Following the concept of statistical convergence and statistical cluster points of a sequence x x , we give a definition of statistical limit superior and inferior which yields natural relationships among these ideas: e.g., x x is statistically convergent if and only if st - liminf x = st - limsup x \textrm {st}\text {-}\textrm {liminf} x= \textrm {st}\text {-}\textrm {limsup} x . The statistical core of x 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 C_{1} -summable to its statistical limit superior is statistically convergent.