Let (left{X_{n}ight}_{n=1}^{infty}) be a sequence of independent random variables where (X_{n}) has a (mathrm{N}left(0, n^{-1}ight)) distribution for

Question:

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

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