Averaging sequences, deranged mappings, and a problem of Lampert and Slater
For a given triangular array \(p_{n,k}\), \(0\leq k\leq n\) and \(n\geq 1\), of probabilities whose row sums equal 1, and six positive constants \(\alpha\), \(\beta\), \(a\), \(b\), \(\lambda\), \(\phi\), let the sequence \(q_n\) be defined as follows: \(q_0= 0\), \(q_1= 1\), and \(q_{n+1}= \sum^n_{k=0} p_{n,k} q_k\) for \(n\geq 1\). Define \[ F(X)= a+ b\cos(\lambda X+\phi)\quad\text{and}\quad \Delta(I,J]= \max\{|q_k- F(\log k)|: I< k\leq J\}. \] For the given array \(p_{n,k}\), let \(\mu\) be the mean \(\sum_k kp_{n,k}\) of the \(n\)th row, \(\sigma^2\) be the variance \(\sum_k(k- \mu)^2 p_{n,k}\) of the \(n\)th row, and \({\mathfrak R}_n\) be the set of the \(k\) such that \(|k-\mu|\leq (n+ 1)^{2/3}\). Assume that the following conditions hold: \(\mu= (n+1)\alpha(1+ \theta/n)\), \(\theta\) stands for a real number in the interval \([-1,1]\); \[ \lambda\log\alpha= -2\pi;\;\sigma^2< \beta(n+ 1)+ 1\quad\text{for }n\geq 2;\quad\text{and}\quad \sum_{k\not\in{\mathfrak R}_n} p_{n,k}\leq 2e^{-n^{1/3}/2}. \] Next choose \(N\) so large and \(\delta\) positive but so small that \(N\geq \max((3/\alpha+ 0.01)^3,90/\beta,1000)\) and \(\alpha(1+ \delta+ N^{-1})(1+ N^{-1})(1+ N^{-1}+ \alpha^{-1} N^{-1/3})= 1.\) Let \(c= b\lambda(0.75\lambda+ 0.65)\). Further choose constants \(C_1= 1.02c\beta\alpha^{- 2}+ b\lambda\) and \(C_2= 2(1+|a|+ b+m(b\lambda+ cm)),\) where \(m= \max(1, 1.0001\alpha^{-1}- 1)\). Let \(I\) be the largest integer possible, subject to \(\alpha N(1- N^{-1}- \alpha^{-1} N^{-1/3})\geq I+ 1\). The authors prove that, for every \(n> I\), \[ |q_n- F(\log n)|\leq\Delta (I,N]+ C_1N^{-1}(1+ \delta^{-1})+ C_2 e^{- N^{1/3}/2}/(1- e^{-\delta N^{1/3}/6}). \] In particular, if the latter is less than \(b\), then the sequence \(q_n\) has no limit. This theorem provides a general setting to answer a question posed by \textit{D. E. Lampert} and \textit{P. J. Slater} [Am. Math. Mon. 105, No. 6, 556-558 (1998)].
scientific article; zbMATH DE number 1525306
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