On the discounted global CLT for some weakly dependent random variables

Abstract

In this paper, we consider L ₁ upper bounds in the global central limit theorem for the sequence of r.v.’s (not necessarily stationary) satisfying the ψ-mixing condition. In a particular case, under the finiteness of the third absolute moments of summands A _i and that of the series ∑_r⩾1 r ² φ(r), we obtain bounds of order O(n ^−1/2) for Δ_n1:= ∫ ^∞_−∞ |ℙ{A ₁ + ⋯ + A _n < x} − Φ(x)|dx, where $A_i = X_i /B_n , B_n^2 = \mathbb{E}(X_1 + \cdots + X_n )^2 > 0,\Phi (x)$ is the standard normal distribution function, and ψ is the function participating in the definition of the ψ-mixing condition. Moreover, we apply the obtained results to get the convergence rate in the so-called discounted global CLT for a sequence of r.v.’s, satisfying the ψ-mixing condition. The bounds obtained provide convergence rates in the discounted global CLT of the same order as in the case of i.i.d. summands with a finite third absolute moment, i.e., of order O((1 − υ)^1/2), where υ is a discount factor, 0 < υ < 1.