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Probability density function of normality is unbounded. How can this be?
For a vector $x$ in $\mathbb{R}^d$, where $d \in \mathbb{N}$, I’m told that the probability density function associated with the normality with location zero and scale $\sigma$ is
$$f(x) = \frac{1}{2\sigma \sqrt{(2\pi)^d}} \exp \left( – \frac{1}{2\sigma^2} \| x \|^2 \right)$$
where $\| \cdot \|$ is the vector 2-norm of vectors in $\mathbb{R}^d$. Since $f(x)$ is unbounded as $x \to \infty$, does that mean that $f$ is not a valid probability density function?
The last part of this statement does not make sense to me; a statement that the pdf is unbounded is not a probabilistic statement.
A:
You’re assuming that the support of $f(x)$ is $(\mathbb{R}^d,\|\cdot\|)$, while the domain of definition of the pdf is $(\mathbb{R}^d,\|\cdot\|_2)$, where $\|x\|_2 = (\sum_i x_i^2)^{1/2}$. The normality pdf is not defined in the $x_i$ of zero sum, so in that case $f$ is not defined.
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