Is convergence in probability implied by consistency of an estimator?

Question

Every definition of consistency I see mentions something convergence in probability-like in its explanation.

From Wikipedia's definition of consistent estimators:

having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to $θ_0$.

That sounds like consistency implies convergence in probability. Though, I never see it said so plainly.

Likewise, from this discussion of unbiasedness and consistency:

Roughly speaking, consistency means that for large values of 𝑛 we are going to be close to the true value of the parameter with a high probability

Is it the case that estimator consistency $\iff$ convergence in probability? Consistency seems somewhat informally defined, and I'm not seeing it stated mathematically.

Answer 1

"Consistency $\iff$ convergence in probability" is what I was taught. Formally (definition from Rice's Mathematical Statistics & Data Analysis, Ch. 8.4, 3rd edition):

"Let $\hat{\theta}_n$ be an estimate of a parameter $\theta$ based on a sample of size $n$. Then $\hat{\theta}_n$ is said to be consistent in probability if $\hat{\theta}_n$ converges in probability to θ as $n$ approaches infinity; that is, for any $\epsilon > 0$, $P(|\hat{\theta}_n − \theta| > \epsilon) \to 0$ as $n\to\infty$."

Rice abbreviates "consistent in probability" (from the above definition) to "consistency" for the rest of the section.

My old lecture notes (notes from an older offering here) are even more blunt: a consistent estimator is by definition one for which $\hat{\theta}_n \overset{p}{\to} \theta$.

So I think that, in general, "consistency $\iff$ convergence in probability" is a fair characterization. In any case, the above definitions are the sense in which I hear "consistency" used most often, and maps on to weak consistency as defined by Amemiya & Takeshi (as noted above).