On the maximum of covariance estimators

Let {X"k,[email protected]?Z} be a stationary process with mean 0 and finite variances, let @f"h=E(X"kX"k"+"h) be the covariance function and @[email protected]^"n","[email protected]?"i"="h"+"1^nX"iX"i"-"h its usual estimator. Under mild weak dependence conditions, the distribution of the vector (@[email protected]^"n","1,...,@[email protected]^"n","d) is known to be asymptotically Gaussian for any [email protected]?N, a result having important statistical consequences. Statistical inference requires also determining the asymptotic distribution of the vector (@[email protected]^"n","1,...,@[email protected]^"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 {@[email protected]^"n","[email protected]"h,[email protected][email protected]?d"n} for some sequences d"n->~ and used these to construct simultaneous confidence bands for @[email protected]^"n","h, [email protected][email protected]?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 (@[email protected]^"n","1,...,@[email protected]^"n","d) and prove also that the limiting distribution of max"1"@?"h"@?"d"""n|@[email protected]^"n","[email protected]"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 Cramer-Wold device, which has some interest in itself.