A new maximal inequality and invariance principle for stationary sequences

We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe (2000). 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. Short title: A New Maximal Inequality I Results Let (Xi)i∈Z be a stationary sequence of centered random variables with finite second moment (E[X 1 ] <∞ and E[X1] = 0). Denote by Fk the σ–field generated by Xi with indices i ≤ k , and define Sn = n ∑ i=1 Xi , Wn(t) = S[nt] √ n , 0 ≤ t ≤ 1 where ‖X‖ = √ E(X2) and [x] denotes the integer part of x. Finally, let W = {W (t) : 0 ≤ t ≤ 1} be a standard Brownian motion. In the sequel =⇒ denotes the weak convergence. Theorem 1 Assume that ∞ ∑ n=1 ‖E(Sn|F0)‖ n3/2 <∞ . (1) Then, {max1≤k≤n S k/n : n ≥ 1} is uniformly integrable and Wn(t) =⇒ √ ηW (t) , where η is a non-negative random variable with finite mean E[η] = σ and independent of {W (t); t ≥ 0}. Moreover, Condition (1) allows to identify the variable η from the existence of the following limit lim n→∞ E(S n|I) n = η in L1 (2) where I is the invariant sigma field. In particular, limn→∞E(S n)/n = σ. In the next theorem we show that, in its generality, condition (1) is optimal in the following sense. Mathematical Subject Classification (2000):60F05, 60F17