Convergence rates and convergence-order profiles for sequences (Q1174195)

The authors investigate a number of methods of measuring convergence rates of sequences and clarify the relationships between these different measures of convergence rates. The authors give eight such measurement methods and demonstrate relations between them. We formulate one of the statements of the present paper. Let us consider a monotone increasing sequence \(\{\lambda_ n\}\) of real numbers converging to the limit \(\lambda^*\). The \(Q\) convergence rate of \(\{\lambda_ n\}\) \((Q)\) is defined as follows: Put \(Q_ p(n):=|\lambda^*-\lambda_ n|/|\lambda^*-\lambda_{n- 1}|^ p(*)\), \(p_ 1:=\inf\{p\mid \lim_{n\to\infty}\sup Q_ p(n)=\infty\}\), \(p_ u:=\sup\{p\mid \lim_{n\to\infty}\inf Q_ p(n)=0\}\). The \(Q\)-convergence rate of \(\{\lambda_ n\}\) is \(p\), if \(P=P_ 1=P_ u\). The \(C\)-convergence rate of \(\{\lambda_ n\}\) is \(P\), if \(\lim_{n\to\infty}Q_ p(n)=C_ p\), with \(0<C_ p<+\infty\) for \(p>1\) and \(0<c_ 1<1\), for \(p=1\). \(Q'\) and \(C'\)-convergence rates are defined analogously, but with the substitutions of \(|\lambda_{j+1}- \lambda_ j|\) for \(|\lambda^*-\lambda_ j|\) \((j=n-1,n)\) made in \((*)\). Theorem 2.2. \(Q'\)-order \(p\) implies \(C'\)-order \(p\), and \(C\)-order \(p\) implies \(Q\)-order \(p\).