Rates of convergence for classes of functions: The non-i.i.d. case
Let \(\{X_ i:\) \(i\geq 1\}\) be a sequence of \(\phi\)-mixing random variables taking values in a sample space (X,A). Let \(G_ n\), \(n\geq 1\), be classes of real-valued measurable functions, \(g: X\to {\mathbb{R}}\). For such \(g\in G_ n\), define \[ S_ n(g)=\sum^{n}_{i=1}\{g(X_ i)- Eg(X_ i)\}. \] This paper establishes bounds on the fluctuations of \(S_ n(g)\) over \(g\in G_ n\) and \(n\geq 1\). Almost surely, with respect to the outer probability measure, \[ \overline{\lim}_{n\to \infty}\sup_{g\in G_ n}| H^{-1}(nV)^{-1/2}S_ n(g)| \leq U<+\infty. \] Here V is the supremum of \(Eg^ 2(X_ i)\) over \(g\in G_ n\) and \(i\leq n\). \(H^ 2\) is the entropy of the class \(G_ n\). These results require that the \(\phi\)-mixing coefficients of \(\{X_ i:i\geq 1\}\) be summable, that \(H(V/n)^{1/2}\) be bounded or convergent to 0, and that the entropies of the classes \(G_ n\) be suitably controlled. The proofs combine the techniques of metric entropy with bracketing, generalizations of inequalities due to Bernstein and Ottaviani to \(\phi\)- mixing sequences, and blocking techniques for \(\phi\)-mixing sequences.
scientific article; zbMATH DE number 3975995