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 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)].