# Asymptotics is strange...

It is a useful trick to “flip” the denominator into numerator when it comes to proving asymptotic properties of errors. Here’s how to do so. Consider a form

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# [Real Analysis] Ch 4. Hilbert Space Theory

# [Real Analysis] Ch 2. Construction of Lebesgue Measure

In this chapter, we construct the Lebesgue measure on $\mathbb{R}^d.$ For this, we prove Riesz representation theorem and use the result to construct a complete measure space $(\mathbb{R}, \mathfrak{M}, m)$ such that integration with respect to $m$ is equal to Riemann integration for all Riemann-integrable functions. We then use $\sigma$-compactness of $\mathbb{R}$ to show that such $m$ is the Lebesgure measure.

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# [Nonparametric] 4. Multivariate Kernel Density Estimation

Until now, I only covered univariate cases. From now on, our focus is on multivariate cases where samples $X_1,\cdots,X_n\in\mathbb{R}^d$ are independently drawn from the density $p:\mathbb{R}^d\to[0,\infty)$ with respect to the Lebesgue measure.

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# [Nonparametric] 3. Bandwidth Selection

Recall that the optimal bandwidth $h_\text{opt}$ derived from MISE was given as the following:

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