# Consistency of M-estimators

Let $\Theta \subseteq \mathbb{R}^{p}$ be compact. Let $Q: \Theta \rightarrow \mathbb{R}$ be a continuous, non-random function that has a unique minimizer $\theta_{0} \in \Theta$.

Let $Q_{n}: \Theta \rightarrow \mathbb{R}$ be any sequence of random functions such that

$$\sup_{\theta \in \Theta} |Q_{n}(\theta) - Q(\theta)| \rightarrow 0$$ as $n \rightarrow \infty$.

If $\theta_{n}$ is any sequence of minimizers of $Q_{n}$, then $\hat \theta_{n} \rightarrow \theta_{0}$ in probability as $n \rightarrow \infty$.

• statistics