Prediction of weakly stationary sequences on polynomial hypergroups
Let \((R_n)\) be a fixed polynomial sequence that induces a polynomial hypergroup on \({\mathbb N}_0\), i.e. that degree\((R_n)=n\) and that the sequence is orthogonal with respect to a probability measure \(\pi\in M^1({\mathbb R})\). Assume that \(R_n(1)=1\) and that all the coefficients \(g(m,n;k)\) defined by \(R_m(x) R_n(x) = \sum_{k=|n-m|}^{n+m} g(m,n;k) R_k(x)\) are nonnegative. A sequence \((X_n)\) of square integrable random variables is weakly stationary on the polynomial hypergroup if \(d(m,n)=E(X_m \overline X_n)\) is bounded and fulfills \(d(m,n)= \sum_{k=|n-m |}^{n+m} g(m,n;k) d(k,0)\). The authors present examples of such sequences. Let \(\widehat X_{n+1}\) be the best linear prediction of \(X_{n+1}\) based on \(X_0,\dots, X_n\). If \(\delta_n=\|\widehat X_{n+1} -X_{n+1}\|_2\to 0\) as \(n\to\infty\), then \((X_n)\) is called asymptotically \(R_n\)-deterministic. For many cases the conditions for \(\delta\to 0\) are given including the respective convergence orders.
scientific article; zbMATH DE number 1906116
scientific article