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For any compact sets $A_1, A_2 \\subset \\mathbb{R}^N$, precise asymptotic behavior of the excursion probability \\[ \\mathbb{P}\\bigg(\\max_{s\\in A_1} X_1(s)>u,\\, \\max_{t\\in A_2} X_2(t)>u\\bigg),\\ \\ \\text{ as }\\ u \\rightarrow \\infty \\] is investigated by applying the double sum method. 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For any compact sets $A_1, A_2 \\subset \\mathbb{R}^N$, precise asymptotic behavior of the excursion probability \\[ \\mathbb{P}\\bigg(\\max_{s\\in A_1} X_1(s)>u,\\, \\max_{t\\in A_2} X_2(t)>u\\bigg),\\ \\ \\text{ as }\\ u \\rightarrow \\infty \\] is investigated by applying the double sum method. 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