Limit theorems for functionals of mixing processes with applications to \(U\)-statistics and dimension estimation (Q2731944)

Let \((Z_n)_{n\in\mathbb{Z}}\) be a stationary, absolutely regular stochastic process, defined on a probability space \((\Omega,\mathbb{F},P)\). Hence \((Z_n)_{n\in\mathbb{Z}}\) satisfies the condition NEWLINE\[NEWLINE\beta_k=2 \sup_n\left \{\sup_{A\in {\mathcal A}^\infty_{n+k}} \biggl(P\bigl( A\mid{\mathcal A}_1^n \bigr)-P(A)\biggr) \right\} \to 0NEWLINE\]NEWLINE where \({\mathcal A}_n^m= \sigma(Z_n, Z_{n+1}, \dots,Z_m)\) for \(m\geq n\). We call a sequence \((X_n)_{n\in\mathbb{Z}}\) a two-sided functional of \((Z_n)_{n\in\mathbb{Z}}\) if there is a measurable function \(f\) defined on \(\mathbb{R}^\mathbb{Z}\) such that \(X_n=f( (Z_{n+k})_{k\in\mathbb{Z}})\). The authors investigate the asymptotic properties of the sequence \((X_n)_{n\in\mathbb{Z}}\) of such functionals. Specifically, they prove some moment inequalities, which enable us to prove several limit theorems, and obtain the asymptotic properties of \(U\)-statistics and \(U\)-processes (indexed by some class of functions). Using the results, they study the asymptotic distributions of estimators of the fractal dimension of the attractor of a dynamical system. Many examples are also shown.