Extreme value theory for triangular arrays of dependent random variables

The central result of extreme value theory proved by Gnedenko [1] classifies all types of asymptotic distributions that the normalized maximum of a sample of independent identically distributed random variables could possibly have. Does a similar result hold if the variables are not identically distributed or are dependent? We consider a sequence of random vectors Xn = (X1,n, . . . , Xd,n) ∈ R, where d = d(n) ∈ N and n ∈ N. Let [d] := {1, . . . , d}. If, for any fixed x ∈ R, ∣∣∣∣P(max i∈[d] Xi,n ⩽ x)− ∏ i∈[d] P(Xi,n ⩽ x) ∣∣∣∣ → 0, as n →∞, (1)