On normal approximation for φ-mixing and m-dependent random variables

Abstract

In this paper, we estimate the difference |Eh(Z_n) − Eh(Y)| between the expectations of real finite Lipschitz function h of the sum Z_n = (X₁ + ⋯ + X_n)/B_n, where \({B}_{n}^{2}\) = E(X₁ + ⋯ + X_n)² > 0, and a standard normal random variable Y, where real centered random variables X₁,X₂,… satisfy the φ-mixing condition, defined between the “past” and “ future”, or are m-dependent. In particular cases, under the condition \({\sum }_{r=1}^{\infty }r\varphi (r)<\infty \) or \({\sum }_{r=1}^{\infty }{r\varphi }^{1/2}(r)<\infty \), the obtained upper bounds for φ-mixing random variables are of order O(n^−1/2). In addition, we refine the previously known upper bounds of order O((m + 1)^1+δL_2+δ,n), where L_2+δ,n is the Lyapunov fraction of order 2 + δ, for m-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.