About the Lindeberg method for strongly mixing sequences

Abstract

Abstract. We extend the Lindeberg method for the central limit theorem to strongly mixing sequences. Here we obtain a generalization of the central limit theorem of Doukhan, Massart and Rio to nonstationary strongly mixing triangular arrays. The method also provides estimates of the L evy distance between the distribution of the normalized sum and the standard normal.

Central limit theorem for sampled sums of dependent random variables

“…However, they assume that the random variables ξ i are uniformly bounded. Their proof is a variation of the Lindeberg method developed in Rio [21]. Also using a variation of this method, Bardet et al [2] prove a triangular central limit theorem, requiring moments of order 2 + δ, δ > 0.…”

Rates of Convergence in the CLT for Some Weakly Dependent Random Variables

“…We do not know whether this rate is optimal, but we note that also for bounded mixing sequences, Rio [21] obtained the convergence rate 0(n -1 /…”

“…Notice that such sequences are special weakly depen dent sequences as defined in [11]. This implies that nontrivial adaptations are necessary here to adapt Rio's [21] method (for strongly mixing sequences) in Lindeberg theorem.…”

“…Adapting the Lindeberg method, Rio [22] obtained the optimal Berry-Esseen rate for a class of weakly depen dent r.v. 's (see also [21]). Recently, Dedecker [8] uses this approach to prove a CLT for stationary random fields under a projective criterion.…”

“…For (AG) sequences bounded by M, we obtain a convergence rate 0(n"*/ 3 ), if 0 < 6 < 1 and if [19] also [20]). In particu lar, we obtain for nondegenerate sequences of centered and associated r.v.…”