On the maximum of covariance estimators

Let {Xk,k∈Z}{Xk,k∈Z} be a stationary process with mean 0 and finite variances, let ϕh=E(XkXk+h)ϕh=E(XkXk+h) be the covariance function and ϕ̂n,h=1n∑i=h+1nXiXi−h its usual estimator. Under mild weak dependence conditions, the distribution of the vector (ϕ̂n,1,…,ϕ̂n,d) is known to be asymptotically Gaussian for any d∈Nd∈N, a result having important statistical consequences. Statistical inference requires also determining the asymptotic distribution of the vector (ϕ̂n,1,…,ϕ̂n,d) for suitable d=dn→∞d=dn→∞, but very few results exist in this case. Recently, Wu (2009) [19] obtained tail estimates for the vector {ϕ̂n,h−ϕh,1≤h≤dn} for some sequences dn→∞dn→∞ and used these to construct simultaneous confidence bands for ϕ̂n,h, 1≤h≤dn1≤h≤dn. In this paper we prove, for linear processes XnXn and for dndn growing with at most logarithmic speed, the asymptotic joint normality of (ϕ̂n,1,…,ϕ̂n,d) and prove also that the limiting distribution of max1≤h≤dn|ϕ̂n,h−ϕh| is the Gumbel distribution exp(−e−x)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.