Lithuanian Mathematical Journal

, Volume 47, Issue 3, pp 327–335

# On normal approximation of discounted and strongly mixing random variables

• J. Sunklodas
Article

## Abstract

We estimate the difference $$\left| {\mathbb{E}h(Z_v ) - \mathbb{E}h(N)} \right|$$ for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z v = B v −1 i=0 v i X i and $$B_v^2 = \mathbb{E}(\sum\nolimits_{i = 0}^\infty {\upsilon ^i X_i } )^2 > 0$$ with discount factor ν such that 0 < ν < 1. Here {X n , n ≥ 0} is a sequence of strongly mixing random variables with $$\mathbb{E}X_n = 0$$, and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)1/2).

## Keywords

discounted central limit theorem strong mixing condition Lipschitz condition Stein’s method

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