Are these estimators consistent?

Question

Let $\Theta \in \mathbb{R}^d$ be a compact set and $\rho(x,\theta):\mathbb{R}^p\times\Theta\to\mathbb{R}^+$ a continuous function.

The estimator is $\displaystyle \hat{\theta} = \operatorname*{argmax}_{v\in\Theta} \frac{1}{n}\sum_{i=1}^n \rho (X_i,v)$ and I'm trying to determine if it is consistent. I know the estimator is consistent if it converges in probability to $\theta,$ but I fail to see an answer from the given hypothesis, either positive or negative.

Answer 1

I believe this is an M Estimator as specified here, of p-type.

Page 7 of these notes gives sufficient conditions for an M-estimator to be consistent, using their notation:

1. $\sup_{\theta\in\Theta} |M_n(\theta)-M(\theta)| \overset{P}{\rightarrow} 0$
2. $\forall \epsilon>0$ $\sup\{M(\theta): d(\theta,\theta_0)\geq \epsilon\} <M(\theta_0)$
3. $M_n(\hat{\theta}_n) \geq M_n(\theta_0) - o_P(1)$

Then $\hat{\theta}_n \overset{P}{\rightarrow} \theta$

I'm trying to show that given the information in the question, but I'm not able to do it easily. There are some other notes giving sufficient conditions, but the question is kind of sparse on information.