I am having difficulties finding the answer to this question online. I feel like my intuition may mislead me on this problem, and I want a sanity check. This is a simple example for a more complicated issue I'm facing in my research and my desire to not utilize a leave one out estimator.
Assume we have a sequence of IID random samples with mean $\mu$: \begin{align*} X_1, X_2, ..., X_N \end{align*} where $X_i \in \mathbb{R}$. The sample mean is given by: \begin{align*} \mu_N = \frac{1}{N}\sum_{i=1}^{N} X_i \end{align*}
Let $I_A$ and $I_B$ be arbitrary subsets of $\mathbb{R}$. I want to know if $\mu_N$ is asymptotically independent of any $X_k$ for $k \in \left\{1,...,N\right\}$: \begin{align*} \lim_{N \rightarrow \infty}\left|P\left(\mu_N\in I_A, X_k \in I_B\right) - P\left(\mu_N \in I_A \right) P\left(X_k \in I_B\right)\right| = 0 \end{align*} My intuition says they are asymptotically independent. Informally speaking, $X_k$ has less and less influence over $\mu_N$ as $N$ grows large, but maybe this notion of convergence is too strong? The use of leave one out estimators in consistency arguments and the difficulty I experienced finding an answer to this question makes me think that $\mu_n$ and $X_k$ are not asymptotically independent.
Hopefully this is obviously wrong or right to someone! If it is correct, could you comment on how this extends to other estimators fit on IID samples? If it isn't correct, is there a weaker statement that is true?
Thanks in advance.