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Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = \frac{1}{n}\sum_{i=1}^n K_h (x - x_i) \quad = \frac{1}{nh} \sum_{i=1}^n K\Big(\frac{x-x_i}{h}\Big), \end{align} where $K$ a symmetric non-negative function that integrates to one.

I am interested what happens in the limit $n \to \infty$. Are there any publications that prove \begin{align} \hat{f}_h(x) &= \int K_h (x - x_i) \text{d}x \quad \text{or} \\ &= \int K_h (x - x_i) \text{d}P ? \end{align} Is then in the limit $\hat{f}_h(x) = f$?

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You are asking about the consistency of kernel density estimates. See this paper to shows that such estimators are generally consistent under modest conditions.

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