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Given an analytic curve $\\phi: I=[a,b] \\rightarrow H$, we will show that if $\\phi$ satisfies certain geometric condition, then for a typical diagonal subgroup $A =\\{a(t): t \\in \\mathbb{R}\\} \\subset H$ the translates $\\{a(t)\\phi(I)x: t >0\\}$ of the curve $\\phi(I)x$ will tend to be equidistributed in $X$ as $t \\rightarrow +\\infty$. 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