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For~$M_n^{(q)}$ denoting the $q$-th largest value in $\\{X_k : 1\\leqslant k \\leqslant n\\}$, we prove that \\begin{equation*} \\sup_{x\\in\\mathbb{R}} \\left|P\\left(M^{(q)}_n\\leqslant x\\right) - G(x)^n \\sum_{k=0}^{q-1}\\frac{\\left(-\\log G(x)^n\\right)^k}{k!}\\gamma_{q,k}(x)\\right| \\to 0,\\quad \\text{as} \\quad n\\to\\infty, \\end{equation*} for $G$ and $\\gamma_{q,k}$ expressed in terms of maxima over the cycle. 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For~$M_n^{(q)}$ denoting the $q$-th largest value in $\\{X_k : 1\\leqslant k \\leqslant n\\}$, we prove that \\begin{equation*} \\sup_{x\\in\\mathbb{R}} \\left|P\\left(M^{(q)}_n\\leqslant x\\right) - G(x)^n \\sum_{k=0}^{q-1}\\frac{\\left(-\\log G(x)^n\\right)^k}{k!}\\gamma_{q,k}(x)\\right| \\to 0,\\quad \\text{as} \\quad n\\to\\infty, \\end{equation*} for $G$ and $\\gamma_{q,k}$ expressed in terms of maxima over the cycle. 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