# Conjectures

On this page, I occasionally list some conjectures that appeared in my papers. If you are able to prived a solution, please let me know.

## Maximum Number of Oscillations of a Gaussian Convolution

Consider a scalar random variable $$Y=X+Z$$ where $$X$$ and $$Z$$ are independent. Assume that $$Z$$ is a standard normal, and $$X$$ is a bounded symmetric random variable such that $$|X| \le A$$ for some $$A>0$$. Denote by $$f_Y$$ the probability density function of $$Y$$. For obvious reasons, we refer to $$f_Y$$ as a Gaussian convolution. Now denote by $$\mathrm{N}(S,f_Y-\kappa)$$ the number of zeros of a function $$f_Y-\kappa$$ on the set $$S \subseteq \mathbb{R}$$. The conjecture can now be stated as follows:

$\max_{ X: |X| \le A } \max_{ \kappa: \kappa>0} \mathrm{N}( \mathbb{R},f_Y-\kappa)= O(A) \quad (*).$

The first maximization in $$(*)$$ is over all symmetric distributions on $$X$$ that have support on the interval $$[-A,A]$$.

For more details on this conjecture, motivation and supporting arguments see this paper.