Examples concerning Abel and Cesàro limits (Q401107)

The lower and upper Cesàro limits of a sequence \((u_n)_{n=1}^\infty\) are defined, respectively, by NEWLINE\[NEWLINE\underline{C}=\liminf_{n\to\infty}\frac{1}{n} \sum_{i=0}^{n-1}u_i,\quad\overline{C}=\limsup_{n \to \infty}\frac{1}{n} \sum_{i=0}^{n-1}u_i.NEWLINE\]NEWLINE The lower and upper Abel limits of a sequence \((u_n)_{n=1}^\infty\) are defined, respectively, by NEWLINE\[NEWLINE\underline{A}=\liminf_{x\to 1^-}(1-x) \sum_{n=0}^{\infty}u_n x^n,\quad\overline{A}=\limsup_{x \to 1^-}(1-x) \sum_{n=0}^{\infty}u_n x^n.NEWLINE\]NEWLINENEWLINENEWLINEFor a sequence bounded above or below, the following inequalities hold: NEWLINE\[NEWLINE\underline{C}\leq \underline{A}\leq\overline{A}\leq\overline{C}. \eqno{(*)}NEWLINE\]NEWLINENEWLINENEWLINEIf \(\underline{A}=\overline{A}\), then NEWLINE\[NEWLINE\underline{C}=\underline{A}=\overline{A}=\overline{C}. \tag{1}NEWLINE\]NEWLINENEWLINENEWLINEAccording to the authors, either the equalities (1) are satisfied or only the following relations are possible: NEWLINE\[NEWLINE\underline{C}<\underline{A}<\overline{A}<\overline{C},\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\underline{C}=\underline{A}<\overline{A}=\overline{C},\tag{3}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\underline{C}<\underline{A}<\overline{A}=\overline{C},\tag{4}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\underline{C}=\underline{A}<\overline{A}<\overline{C}.\tag{5}NEWLINE\]NEWLINE For the inequalities (2), a number of authors provided some examples of bounded sequences. In this paper, the authors describe examples for which the inequalities (3), (4) and (5) hold for bounded sequences. They also discuss the application of Tauberian and Hardy-Littlewood theorems to Markov decision processes.