On the maximum of covariance estimators

Let { X k , k ∈ Z } be a stationary process with mean 0 and finite variances, let ϕ h = E ( X k X k + h ) be the covariance function and ϕ ̂ n , h = 1 n ∑ i = h + 1 n X i X i − h its usual estimator. Under mild weak dependence conditions, the distribution of the vector ( ϕ ̂ n , 1 , … , ϕ ̂ n , d ) is known to be asymptotically Gaussian for any d ∈ N , a result having important statistical consequences. Statistical inference requires also determining the asymptotic distribution of the vector ( ϕ ̂ n , 1 , … , ϕ ̂ n , d ) for suitable d = d n → ∞ , but very few results exist in this case. Recently, Wu (2009) [19] obtained tail estimates for the vector { ϕ ̂ n , h − ϕ h , 1 ≤ h ≤ d n } for some sequences d n → ∞ and used these to construct simultaneous confidence bands for ϕ ̂ n , h , 1 ≤ h ≤ d n . In this paper we prove, for linear processes X n and for d n growing with at most logarithmic speed, the asymptotic joint normality of ( ϕ ̂ n , 1 , … , ϕ ̂ n , d ) and prove also that the limiting distribution of max 1 ≤ h ≤ d n | ϕ ̂ n , h − ϕ h | is the Gumbel distribution exp ( − e − x ) . This partially verifies a conjecture of Wu (2009) [19]. The proof is based on a quantitative version of the Cramér-Wold device, which has some interest in itself.