Extremes of projections of functional time series on data-driven basis systems (Q726120)

In this manuscript the authors consider a problem of derivation under what conditions the vectors \(\widehat{\mathbf Z}_i=[\widehat{Z}_{i1},\widehat{Z}_{i2},\dots,\widehat{Z}_{ip}]'\), \(1\leq i\leq N\), have the same asymptotic extreme behavior as the unobservable vectors \(\mathbf Z_i=[Z_{i1},Z_{i2},\dots,Z_{ip}]'\) as \(N\) tends to \(\infty\). The elements \(\widehat{Z}_{ij}\) are the estimated scores and appear in the expansion \(X_i(t)\approx \sum_{j=1}^p \widehat{\lambda}_{j}^{1/2} \widehat{Z}_{ij} \widehat{v}_j(t)\), where \(X_i\) is a function in the space \(L^2\) of a strictly stationary mean zero time series \(\{X_i\}\). The convergence in distribution of extremes of the random variables \(\langle {X_i},{v_j}\rangle\) and \(\langle {X_i},{\widehat{v}_j}\rangle\) are derived under general assumptions and the rates of convergence of their distribution functions are considered.