A strong invariance principle for the extremes of multivariate stationary \(m\)-dependent sequences (Q1200621)

Let \(\{{\mathbf X}_ n=(X_ n^{(1)},\dots,X_ n^{(d)})\), \(n\geq 1\}\) be a strictly stationary \(m\)-dependent sequence of random vectors and satisfying additional conditions on the bivariate distributions. Let \(\{{\mathbf M}_ n=(M_ n^{(1)},\dots,M_ n^{(d)})\), \(n\geq 1\}\), where \(M_ n^{(i)}=\max(X_ 1^{(i)},\dots,X_ n^{(i)})\), \(1\leq i\leq d\), \(d\geq 1\). It is shown that there exists a probability space which carries, in addition to the sequence \(\{{\mathbf X}_ n\), \(n\geq 1\}\), an i.i.d. sequence \(\{\widehat{\mathbf X}_ n\), \(n\geq 1\}\) having for all \(n\geq 1\) independent uniformly on \([0,1]\) distributed components \(X_ n^{(i)}\), \(1\leq i\leq d\), such that if \(\{\widehat{M}_ n^{(i)}:=\max(\widehat X_ 1^{(i)},\dots,\widehat X_ n^{(i)})\), \(1\leq i\leq d\}_{n\geq 1}\), then there exist almost surely an \(n_ 0\) and \(q\) such that for all \(n\geq n_ 0\), \(\widehat{{\mathbf M}}_ n={\mathbf M}_{n+q}\). The obtained result is an extension of the author's result [Ann. Inst. Henri Poincaré, Probab. Stat. 23, 425-457 (1987; Zbl 0631.60031)] to the case \(d>1\).