Dirac delta function of a function of different variable

Question

We know that $\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$. But how do we find $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n{\delta(x-t_{i})} $$ if $t_{i}$ is an random entry of a Gaussian random variable?

Answer 1

Let's solve the problem for the finite case and then apply the limits. Let $t_{i}$s be the random entries such that the value $T_{i}$ is assumed $n_{i}$ times.

Now, $$\frac{1}{n}\sum_{i=1}^n{f(t_{i}))}=\frac{1}{n}\sum_{j=1}^m{n_{j}f(T_{j}))}$$ $$=\sum_{j=1}^m{\frac{1}{n}{n_{j}}f(T_{j})}=\sum_{j=1}^m{p_{j}f(T_{j})}$$

Taking the continuum limit, $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n{f(t_{i})}=\int\limits_{-\infty}^{\infty}p(T)f(T)dT$$

Here, $p(t)=\frac{e^{-\frac{t^2}{2}}}{\sqrt{2\pi}}$ and $f(t)=\delta(x-t)$ giving the Gaussian distribution as the required answer.

-My professor's solution