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I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf."

Ex: $X_1,X_2,\ldots$ is a sequence of independent RVs with mean $a$ and variance $\sigma^2$, then the distribution function $F_n(x)$ of the normalized sum $Z_n=\frac{\sum_{i=1}^n (X_i-a)}{\sigma \sqrt n}$ converges to the normal distribution function. However, this does not imply the convergence of density of $p_n(x)$ of $Z_n$ to $\frac{e^{-x^2/2}}{\sqrt{2\pi}}.$

What conditions can be included to ensure convergence of PDFs/PMFs?

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  • $\begingroup$ When you say "convergence", you are specifically looking for pointwise convergence? $\endgroup$ – Nate Eldredge Nov 9 '16 at 14:17
  • $\begingroup$ @NateEldredge, I had the same question - what norms are we looking at?. You can also look at convergence (in some norm) of the CDF. $\endgroup$ – usul Nov 9 '16 at 14:21
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You should look at the old book

Gnedenko and Kolmogorov: Limit Distributions for Sums of Independent Random Variables.

where they have examples of local limit theorems. I assume many more things were discovered in the meantime. Here is an example of theorem that answers your question.

Theorem. Suppose that $(X_n)_{N\geq 1}$ is a sequence of i.i.d. random variables with common density (PDF) $p(x)$. Denote by $p_n(x)$ the density of $Z_n=X_1+\cdots +X_n$. Assume the following conditions.

  • The random variables $X_n$ are $L^2$, i.e., $$ \sigma^2:=\int_{-\infty}^\infty x^2 p(x) dx<\infty. $$
  • There exists $r\in (1,2]$ and a positive integer $n$ such that $\newcommand{\bR}{\mathbb{R}}$ $p_n\in L^r(\bR)$. (This is guaranteed if e.g. $p(x)$ is Lipschitz.)

Then

$$ \sigma\sqrt{n}p_n(\sigma\sqrt{n}\; x) \to \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} $$

uniformly with respect to $x\in\bR$. For a proof and more details see sections 46,47 of the above book.

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  • $\begingroup$ Is it possible to obtain the rate of convergence? I.e. how large is the error/remainder if $n$ is large but not infinite? $\endgroup$ – ManUtdBloke Oct 13 at 11:52
  • $\begingroup$ Have you cehcked Terry Tao's blog on this? terrytao.wordpress.com/2015/11/19/… $\endgroup$ – Liviu Nicolaescu Oct 14 at 14:26
  • $\begingroup$ Terry Tao's blog post seems to only discuss convergence in probability, whereas you have specified pointwise/uniform convergence of the pdf. Pointwise/uniform convergence of the pdf is what I am interested in. $\endgroup$ – ManUtdBloke Oct 14 at 15:16
  • $\begingroup$ In Section 47, p. 228, of the above book of Gnedenko and Kolmogorov there are estimates for the rate of convergence provided $$\int_{-\infty}^\infty |x|^k p(x) dx<\infty,$$ for some $k\geq 3. $ $\endgroup$ – Liviu Nicolaescu Oct 18 at 14:20
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I don't know if this is an answer to your question: Years before I presented the following result in a lecture about weak convergence: Theorem: Assume that $X_n, n \in N$ are real r.v. with unimodal density $\varphi_n$, $P_0$ a probability measure with unimodal, continuous density $\varphi$, unique maximum value $t^*$ and d.f. $\Phi$, Then if $X_n \to P_0$ in distribution, Then $\liminf_{n \to \infty} \inf_{t \in R} (\varphi_n(t) - \varphi(t)) = 0$ $\lim_{n \to \infty} \sup_{|t-t^*| \geq \delta} |\varphi_n(t) - \varphi(t))| = 0$ for any $\delta > 0$ and $\lim_{n \to \infty} \sup_{t \in R} |\varphi_n(t) - \varphi(t))| = 0$ if any $\varphi_n$ is logarithmically concave.

I don't know where you can find this reasult, but the proof is not too hard. There also is a version for discrete r.v.

The usual simple local theorems are special cases.

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