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There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.

Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $f_1, f_2, \dots, f$ be those density functions $\mathbb{R} \to \mathbb{R}$. The convergence of density functions is $\lim_{n\to\infty}f_n = f$ (a.e.) or $\lim_{n\to\infty}\int_\mathbb{R}|f_n-f|=0$.

I think it would be interesting, for example, to prove an analog of the central limit theorem not on cumulative distribution functions but on density functions.

Question: Is there a research on convergence of density functions?

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Such theorems, called local (central) limit theorems, are well known; see e.g. The Encyclopedia of Mathematics.

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  • $\begingroup$ Could you tell me how to prove the theorem?I want to know the proof of the claim in the first half.If $p_{n_0}(x)$ is bounded for some $n_0$, why $p_n(x)\to\frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$ uniformly? $\endgroup$ – Takeo 2 days ago
  • $\begingroup$ @Takeo : A detailed proof of this result can be found e.g. in the book by V.V. Petrov, "Sums of independent random variables" , Springer (1975), referred to in the article in The Encyclopedia of Mathematics linked in the above answer. See Theorem 7 in Section 2 of Chapter VII of Petrov's book. Section 4 of the same chapter contains a large number of refinements of this result, with further references. $\endgroup$ – Iosif Pinelis 2 days ago

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