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where * is the [[convolution]] operator between two functions, and
:<math>\operatorname{Var} \hat{f}(\mathbf{x};\mathbf{H}) = n^{-1} |\mathbf{H}|^{-1/2} R(K)f(\mathbf x) + o(n^{-1} |\mathbf{H}|^{-1/2}).</math>
For these two expressions to be well-defined, we require that all elements of '''H''' tend to 0 and that ''n''<sup>−1</sup> |'''H'''|<sup>−1/2</sup> tends to 0 as ''n'' tends to infinity. Assuming these two conditions, we see that the expected value tends to the true density ''f'' i.e. the kernel density estimator is asymptotically [[Bias of an estimator|unbiased]]; and that the variance tends to zero. Using the standard mean squared value decomposition
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