Given an unbiased estimator $\hat \theta\_n$ of a parameter $\theta$, if the estimator has small variance (approaching $0$ as $n\to\infty$), it seems reasonable to expect that the estimator is consistent (i.e. that $\hat \theta\_n$ converges in probability to the constant $\theta$). 


Is that actually true? 



Is an unbiased estimator with arbitrarily small variance necessarily consistent?
Markov's inequality says that for a non-negative random variable $X$ with expected value $\mu$,
$$
\Pr(X>a) \le \frac \mu a.
$$
(E.g. no more than $1/15$ of the population can have more than $15$ times the average income (assuming all incomes are non-negative), etc.)


So
$$
\Pr(|\hat \theta\_n - \theta| > \varepsilon) = \Pr((\hat\theta\_n-\theta)^2>\varepsilon^2)
\le \frac{\sigma\_n^2}{\varepsilon^2} \to 0\text{ as }n\to\infty
$$
where $\sigma\_n^2=\operatorname{var}(\hat\theta\_n)$.


