For a sequence $\left( X_1, X_2, \dots \right)$ of independent and identically distributed random variables with zero expectation and unit deviation, CLT implies $$ \lim_{n\to\infty} P\left( \frac{1}{\sqrt{n}} \cdot \sum_{i = 1}^n X_i \leq x \right) = \Phi\left( x \right),\quad \forall x \in \mathbb{R}, $$ where $\Phi$ is a CDF of standard normal distribution. To say something about the PDF of the limit of the sequence, I would like to use a derivative and exchange it with the limit like $$ \frac{d}{dx} \Phi\left( x \right) = \underbrace{\frac{d}{dx} \lim_{n\to\infty}} P\left( \frac{1}{\sqrt{n}} \cdot \sum_{i = 1}^n X_i \leq x \right) = \underbrace{\lim_{n\to\infty} \frac{d}{dx}} P\left( \frac{1}{\sqrt{n}} \cdot \sum_{i = 1}^n X_i \leq x \right), \quad \forall x \in \mathbb{R}. $$ However, there are a few issues:
- The probability density function may not exist for any finite $n$: for example, if $X_1$ has a binomial distribution with non-zero probabilities of failure and success and more than one trial. Or is it okay to use Dirac delta function?
- Even if all the densities exist, I need a uniform sequence convergence to exchange a limit sign with a derivative.
Which conditions must be satisfied to infer the PDF from CLT? Does it become more straightforward if the density of $X_1$ exists and is differentiable?