Central limit theorems for conditionally strong mixing and conditionally strictly stationary sequences of random variables (Q6619747)

Let \(X_1,X_2,\ldots\) be a sequence of random variables defined on a probability space \((\Omega,\mathcal{A},P)\) and \(\mathcal{F}\) be a sub-\(\sigma\)-algebra of \(\mathcal{A}\). This sequence is \(\mathcal{F}\)-strictly stationary if the joint distributions of \((X_{n_1},\ldots,X_{n_k})\) and \((X_{n_1+r},\ldots,X_{n_k+r})\), each conditional on \(\mathcal{F}\), are identical for any \(n_1<\cdots<n_k\) and \(r\geq1\). The sequence is \(\mathcal{F}\)-strong mixing if the corresponding conditional strong mixing coefficient converges to zero. The main results of the present paper are central limit theorems for sequences of \(\mathcal{F}\)-strong mixing and \(\mathcal{F}\)-strictly stationary random variables, including under conditions on the rate of convergence of the conditional strong mixing coefficients to zero, and either boundedness or existence of moments of the \(X_i\). Other results of independent interest are also derived, including a conditional covariance bound in terms of conditional quantiles.