Approximation orders in the conditional central limit theorem for weakly dependent random variables (Q1281604)

The author studies the speed of convergence in conditional central limit theorems for weakly dependent sequences. Specifically, assume that \((X_k)_{k\geq 1}\) is strictly stationary and strongly mixing with partial sums \(S_n: =\sum^n_{k=1} X_k\). Then the convergence rate of \(\sup_{t\in\mathbb{R}}|P(S_n/ \sqrt{\text{Var}(S_n)}\leq t\mid B)- \Phi(t) |\) is estimated from above, under conditions on the set \(B\), on the mixing coefficients and on the moments of the process \((X_k)_{k\geq 1}\). The present paper generalizes results of \textit{D. Landers} and \textit{L. Rogge} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 66, 227-244 (1984; Zbl 0525.60029)] for independent random variables and of the author [Vietnam J. Math. 21, 13-19 (1993)] for uniformly mixing processes.