# Proof that a zero-variance Gaussian function becomes a Delta distribution

Consider the Gaussian function:

$$f_{N(\mu, \sigma^2)}(t) = \frac{1}{\sigma \sqrt{\pi}} \exp \left[ {-\frac{1}{2}\left( \frac{t - \mu}{\sigma} \right)^2} \right]$$

I have seen in some texts showing that:

$$\lim_{\sigma \to 0} f_{N(\mu, \sigma^2)}(t) = \delta(t-\mu)$$

Now, I know that the Dirac's Delta has been the subject of many studies and has been defined in many ways. One of the early definitions is that such a function can be modeled as a impulse whose base's length is a function of $$R^{-1}$$ and height is a function of $$R$$ (for a given parameter $$R \in \mathbb{R}$$).

### Distribution theory

However, modern Distribution Theory given this function a full characterization as a distribution (aka generalized function) defined as:

$$\langle \delta, \phi \rangle = \phi(0)$$

Having $$\phi \in \mathcal{D}$$ a test function.

How can we prove, in the distribution framework, that the Gaussian function converges to a Dirac Delta distribution when the variance vanishes?

### Attempt

I have tried setting a proof. Since we need to prove that a function converges to a distribution, we first need to work in the distribution space by considering the generalized function of the Gaussian function.

I know from Distribution Theory that a generic function $$f:\mathbb{R} \mapsto \mathbb{R}$$ can be transformed into a distribution $$T_f$$ by:

$$\langle T_f, \phi \rangle = \int_{-\infty}^{+\infty} f(t)\phi(t)dt$$

So our Gaussian distribution would be the following functional:

$$\langle T_{N(\mu, \sigma^2)}, \phi \rangle = \frac{1}{\sigma \sqrt{\pi}} \int_{-\infty}^{+\infty} \exp \left[ {-\frac{1}{2}\left( \frac{t - \mu}{\sigma} \right)^2} \right] \phi(t) dt$$

From here I am a little insecure how to move on. My goal would be to extract the limit of the distribution and show a convergence to the Delta distribution. I remember that generalized functions have a well defined definition of functional convergence which is based on a convergence model defined on the test function space. How to proceed?

Make the substitution $$s=\frac {t-\mu} {\sigma}$$. You get $$\frac 1 {\sqrt {2\pi}}\int_{\mathbb R} e^{-s^{2}/2} \phi (\mu+\sigma s) ds$$. It is easy to justify taking the limit inside (by DCT) so the expression tends to $$\phi (\mu) \frac 1 {\sqrt {2\pi}}\int_{\mathbb R} e^{-s^{2}/2} ds=\phi (\mu)$$.