For a random variable $X$ with PDF $p(x)$, my goal is to give a convergence rate to $$ \int_\epsilon^{2\epsilon}p(x)dx\overset{\epsilon\to 0}{\to} 0 $$ Note that this is not always the case; e.g. $p(x) = \delta(x)$.

My approach is, find $B$ where $p(x) < B$ for all $x$. Then, this quantity is certainly upper bounded by $B\epsilon$. However, $p(x)$ is super hard to characterize ($X$ is like the sum and product of a lot of things).

1) Are there nice techniques out there used to bound the PDF value of complicated random variables? I feel like with sums I can do something with convolution, but products I'm completely lost.

2) Are there better ways of approaching this goal? I feel like an exponential bound would be excellent, rather than a linear bound. It doesn't quite feel like a concentration inequality would work well here, though.

Thanks for any thoughts!

Edit: Some progress I made. Consider $A$ and $B$ independent random variables, and $C = f(A,B)$. Define $p_A$, $p_B$, $p_C$ the PDFs of $A$, $B$, and $C$, and for any random variable $X$, $p_X^\infty$ the max PDF value. Then $$ p_C(x) = \int_{-\infty}^\infty p_{C|A}(f(A,B) | a) p_A(a) da \overset{(a)}{=}\int_{-\infty}^\infty p_{C|A}(f(A,B)) p_A(a) da\leq p_{C|A}^\infty \int_{-\infty}^\infty p_A(a) = p_{C|A}^\infty $$ So, for example, if $C = (A+B)/2$, then $p_{C|A}^\infty = p_{B/2}^\infty$. Taking $A$ and $B$ Gaussian $\mathcal N(0,1)$, then we can get stuff like $$ p_{A}^\infty = p_B^\infty = \frac{1}{\sqrt{2\pi}\sigma} \Rightarrow p_{B/2}^\infty = \frac{2}{\sqrt{2\pi}\sigma} \leq p_C^\infty $$ This is consistent with what we know about Gaussian random variable behavior.

In other words, if $f$ is a reasonable mixture of independent random variables all with "non-peaky" PDFs, then the PDF of $f$ cannot be peaky.

However, none of this holds if $A$ and $B$ are dependent (step (a) is key). This is now my main challenge; how can we get around this?

  • $\begingroup$ $p(x)=\delta (x)$ doesn't make sense. $\delta$ measures have no densities. $\endgroup$ – Kavi Rama Murthy May 10 '18 at 5:41
  • $\begingroup$ Well, it's a PMF, right? Anyway that example isn't too important since I think the limit is still 0 (which is never achieved.) The issue is more the convergence guarantee itself. Take instead $p(x) = e^{-x+A}$ where the $A$ is just for normalization. Then the convergence is not even linear. $\endgroup$ – Y. S. May 10 '18 at 5:47
  • $\begingroup$ Overall, I have some silly / simple ways of answering this question, but mostly I'm wondering if people have worked on this type of problem before, and have standard techniques that might be more powerful (e.g. concentration inequalities, or something related.) $\endgroup$ – Y. S. May 10 '18 at 5:48
  • $\begingroup$ If by $\delta$ measure you mean a discrete distribution, they indeed have densities - with respect to counting measure. $\endgroup$ – Math1000 May 10 '18 at 6:47
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    $\begingroup$ Ask on the chem and physics stackexchanges. $\endgroup$ – User3910 May 10 '18 at 16:24

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