In the related post over at math.se, the answerer takes as given that the definition for asymptotic unbiasedness is $\lim_{n\to \infty} E(\hat \theta_n-\theta) = 0$.
Intuitively, I disagree: "unbiasedness" is a term we first learn in relation to a distribution (finite sample). It appears then more natural to consider "asymptotic unbiasedness" in relation to an asymptotic distribution. And in fact, this is what Lehmann & Casella in "Theory of Point Estimation (1998, 2nd ed) do, p. 438 Definition 2.1 (simplified notation):
$$\text{If} \;\;\;k_n(\hat \theta_n - \theta )\to_d H$$
for some sequence $k_n$ and for some random variable $H$, the estimator $\hat \theta_n$ is
asymptotically unbiased if the expected value of $H$ is zero.
Given this definition, we can argue that consistency implies asymptotic unbiasedness since
$$\hat \theta_n \to_{p}\theta \implies \hat \theta_n - \theta \to_{p}0 \implies \hat \theta_n - \theta \to_{d}0$$
...and the degenerate distribution that is equal to zero has expected value equal to zero (here the $k_n$ sequence is a sequence of ones).
But I suspect that this is not really useful, it is just a by-product of a definition of asymptotic unbiasedness that allows for degenerate random variables. Essentially we would like to know whether, if we had an expression involving the estimator that converges to a non-degenrate rv, consistency would still imply asymptotic unbiasedness.
Earlier in the book (p. 431 Definition 1.2), the authors call the property $\lim_{n\to \infty} E(\hat \theta_n-\theta) = 0$ as "unbiasedness in the limit", and it does not coincide with asymptotic unbiasedness.
Consistency occurs whenever—
- the estimator is unbiased in the limit, and
- the sequence of estimator variances goes to zero (implying that the variance exists in the first place).
These make up a sufficient, but not necessary condition.
For the intricacies related to concistency with non-zero variance (a bit mind-boggling), visit this post.
