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})

Question:

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 $n ightarrow \infty$.

Fantastic news! We've Found the answer you've been seeking!