# Speed of convergence in probability

Let $X_i$ be a random variable.

Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $E(X_i)=\mu$.

Let $\bar{X}_n:=\frac{1}{n}\sum_{i=1}^{n}X_i$.

Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$

Can we say that $plim_{n \rightarrow \infty}\bar{X}_n=plim_{n \rightarrow \infty}(\bar{X}_n+A_n)=\mu$?

Is there any difference between these two estimators in terms of rate of convergence to $\mu$?