# Convergence requirement of Parzen window estimates

In our Machine Learning class we recently came across Parzen Window Estimates. The following statement was made:

Let $\hat p_n$ be the estimator $\hat p$ using $n$ data points and let $p (x)$ be the real distribution of the data. For an acceptable estimate, we would like to require:

$$\lim_{n \to \infty}{\mathbb{E}[\hat p_n (x)]} = p(x)$$ $$\lim_{n \to \infty}{\text{Var}[\hat p_n (x)]} = 0$$

The first requirement I would paraphrase as 'asymptotically unbiased'. But with the second requirement combined, I don't see the difference to requiring just

$$\lim_{n \to \infty}{\hat p_n(x)} = p(x).$$

Is there an aspect I am missing, or are the two formulations equivalent?

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$\lim_{n\rightarrow\infty}\hat{p}_n(x)=p(x)$ requires that all higher order moments of the distribution converge in the limit, or that they vanish in the limit, whereas the first two statements require only that the first two moments converge.