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Let $\hat{\theta}_N$ be an estimator of $\theta$ constructed from $N$ samples. In the scalar case there is a theorem stating that if the bias and variance of $\hat{\theta}_N$ both go to zero as $N\rightarrow \infty$ then $\hat{\theta }_N$ is a a consistent estimator of $\theta$. Is there a corresponding theorem if $\theta$ is a vector?

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I found my answers in "An Introduction to Multivariate Statistical Analysis", which defines consistency of a vector-estimator as elementwise consistency. This implies that I can apply the scalar theorem elementwise.

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