Let (left{X_{n}ight}_{n=1}^{infty}) be a sequence of independent and identically distributed random variables where (X_{n}=n^{-1} U_{n}) where (U_{n})

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Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables where \(X_{n}=n^{-1} U_{n}\) where \(U_{n}\) has a \(\operatorname{Uniform}(0,1)\) distribution for all \(n \in \mathbb{N}\). Prove that \(X_{n}=o_{p}\left(n^{-1 / 2}ight)\) and that \(X_{n}=O_{p}\left(n^{-1}ight)\) as \(n ightarrow \infty\).

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