A new maximal inequality and invariance principle for stationary sequences

Abstract

We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713-724]. Then, we apply it to establish the Donsker invariance principle for this class of stationary sequences. A Markov chain example is given in order to show the optimality of the conditions imposed.

A maximal 𝕃_{𝕡}-inequality for stationary sequences and its applications

“…We shall first check the convergence part in Proposition 1. Similar to Proposition 3.1 in Peligrad and Utev [16] we note that…”

“…First, we note that ∆ r,p ≤ 9( √ 2+1)δ n,p , which follows from the proof of Lemma 3.3 in Peligrad and Utev [16] applied to the sub-additive sequence V k = E(S k |F 0 ) p for k ≤ n and V k = 0 for k > n. Then, Proposition 1 follows from (10) applied with the inequality…”

“…Motivated by Maxwell and Woodroofe [13], who proved that condition δ ∞,2 < ∞ is enough for the central limit theorem, Peligrad and Utev [16] established the L 2 -maximal inequality and used it to prove the invariance principle.…”

Asymptotic Normality for Weighted Sums of Linear Processes

“…innovations and a j 's such that ∞ j=0 j|a j | < ∞, was established in Phillips and Solo (1992), and for more general stationary processes in Peligrad and Utev (2005).…”

Asymptotic theory for stationary processes

“…Rosenblatt (1956) introduced strong mixing processes, while Gänssler and Häeusler (1979) and Hall and Heyde (1980) considered martingales. For central limit theorems for stationary processes see Ibragimov (1962), Gordin (1969), Ibragimov and Linnik (1971), Gordin and Lifsic (1978), Peligrad (1996), Doukhan (1999), Maxwell and Woodroofe (2000), Rio (2000), Peligrad and Utev (2005), and Bradley (2007).…”