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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 $\mu$?

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Yes, you can. And no, there is no difference (the standard measure of the quality of the estimator is his variance, which does not change when you add something deterministic).

The change between both estimators is than both of them can't be unbiased.

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