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Uniform bound for asymptotic normality of discounted \(m\)-dependent random variables using Heinrich’s method. (English) Zbl 1184.60007

Summary: We estimate the difference \(\left| F_{Z_v} (x) - \Phi (x) \right| \), where \(F_{Z_v} (x) \) is the distribution function of normalized series \(Z_v = B^{-1}_v \sum^{\infty}_{j=0} v^j X_j\) with \(B^{2}_v = \mathbb E (\sum^{\infty}_{j=0} v^j X_j) > 0\) and the discount factor \(v, 0 < v < 1; X_{0},X_{1},X_{2},\cdots \) is a sequence of \(m\)-dependent random variables, and \(\Phi (x)\) is the standard normal distribution function. In a particular case, the obtained upper bound is of order \(O((1 - v)^{1/2})\).

MSC:

60F05 Central limit and other weak theorems
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References:

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