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APPROXIMATION ORDERS IN THE CONDITIONAL CENTRAL LIMIT THEOREM FOR WEAKLY DEPENDENT RANDOM VARIABLES

$icon-email$ BUI KHOI DAM

Abstract

Let $(X_{n})_{n \geq 1}$ be a stationary, strong mixing sequence of random variables with $E X_{n} = 0$ , $E X_{n}^{2} = 1$ and let $B \in σ (X_{1}, X_{2}, \dots, X_{n}, \dots)$ with $P (B) > 0$ . In this note we establish an estimation for the quantity
$Δ_{n} (B) = sup_{t \in R} | P (S_{n} . (E S_{n}^{2})^{- 1 / 2} < t | B) - Φ (t),$
where $Φ (t)$ is a standard normal distribution function and $S_{n} = \sum_{i = 1}^{n} X_{i}$ .

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APPROXIMATION ORDERS IN THE CONDITIONAL CENTRAL LIMIT THEOREM FOR WEAKLY DEPENDENT RANDOM VARIABLES

Abstract