Abstract
Let ξ n be a φ-mixing sequence of real random variables such that \( \mathbb{E}{\xi_n} = 0 \), and let Y be a standard normal random variable. Write S n = ξ 1 + · · · + ξ n and consider the normalized sums Z n = S n /B n , where \( B_n^2 = \mathbb{E}S_n^2 \). Assume that a thrice differentiable function \( h:\mathbb{R} \to \mathbb{R} \) satisfies \( {\sup_{x \in \mathbb{R}}}\left| {{h^s}(x)} \right| < \infty \). We obtain upper bounds for \( {\Delta_n} = \left| {\mathbb{E}h\left( {{Z_n}} \right) - \mathbb{E}h(Y)} \right| \) in terms of Lyapunov fractions with explicit constants (see Theorem 1). In a particular case, the obtained upper bound of Δ n is of order O(n −1/2).
We note that the φ-mixing coefficients φ(r) are defined between the “past” and “future.” To prove the results, we apply the Bentkus approach.
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Kazys Sunklodas, J. Some estimates of the normal approximation for φ-mixing random variables. Lith Math J 51, 260–273 (2011). https://doi.org/10.1007/s10986-011-9124-6
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DOI: https://doi.org/10.1007/s10986-011-9124-6